#499268

order	512
nilpotency class	4
isoclinism	isoclinic class of #499266
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

794 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(5,6,11,16)(7,12,18,34)(8,21,23,56)(9,26,29,61)(10,32,30,15)(13,37,39,24)(14,41,44,31)(17,45,47,55)(19,50,27,60)(20,53,49,36)(22,43,25,59)(28,38,57,40)(33,58,65,48)(35,63,42,52)(46,68,54,66)(51,67,62,64);(1,2)(3,4)(5,7,11,18)(6,12,16,34)(8,17,23,47)(9,27,29,19)(10,20,30,49)(13,33,39,65)(14,42,44,35)(15,36,32,53)(21,55,56,45)(22,54,25,46)(24,58,37,48)(26,60,61,50)(28,62,57,51)(31,63,41,52)(38,67,40,64)(43,68,59,66);(2,4)(5,8)(6,13)(7,17)(9,28)(10,22)(11,23)(12,33)(14,43)(15,38)(16,39)(18,47)(19,51)(20,54)(21,26)(24,31)(25,30)(27,62)(29,57)(32,40)(34,65)(35,66)(36,67)(37,41)(42,68)(44,59)(45,50)(46,49)(48,52)(53,64)(55,60)(56,61)(58,63);(7,18)(8,22,23,25)(9,29)(12,34)(13,38,39,40)(14,44)(17,46,47,54)(20,49)(21,43,56,59)(24,28,37,57)(26,61)(31,41)(33,64,65,67)(36,53)(45,68,55,66)(48,62,58,51)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^b=fi,d^c=dh,e^c=ei,f^c=fh,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281854
maximal normal subgroup of	#72281854

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406464);