#499920

order	512
nilpotency class	4
isoclinism	isoclinic class of #499919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60453	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1368 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1368 conjugacy classes of subgroups — too many to draw. Only its 176 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:

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

Polycyclic presentation:

2^9|c^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407116);