#499985

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

931 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407181);