#512292

order	512
nilpotency class	4
isoclinism	isoclinic class of #511747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

644 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:

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

Polycyclic presentation:

2^9|b^2=efi,c^2=i,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,c^b=ci,g^d=gh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419488);