#508933

order	512
nilpotency class	4
isoclinism	isoclinic class of #508292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13325	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

817 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6)(7,10)(8,9)(11,12)(13,14,17,21,28,34,42,44,16,19,27,33,18,22,30,37)(15,23,25,38,31,43,41,32,24,39,40,35,26,36,29,20);(1,2,3,4)(6,9)(10,12)(14,19)(20,32)(21,33)(22,34)(23,39)(35,38)(36,43)(37,44);(2,4)(7,11)(10,12)(13,15)(14,20)(16,24)(17,29)(18,31)(19,32)(21,36)(22,38)(23,37)(25,30)(26,28)(27,41)(33,43)(34,35)(39,44)(40,42);(5,7,8,11)(6,10,9,12)(15,24)(20,32)(23,39)(25,40)(26,31)(29,41)(35,38)(36,43)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416129);