#506499

order	512
nilpotency class	4
isoclinism	isoclinic class of #501330
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

762 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413695);