#495468

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

586 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=fi,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,h^b=hi,d^c=di

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, 402664);