#494974

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

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	C₂²×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

610 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)(13,14,19,25)(15,21,16,28)(17,23,20,36)(18,32,34,44)(22,27,41,30)(24,31,42,35)(26,37,29,39)(33,40,43,38);(1,3,4,2)(6,9)(10,12)(13,15,19,16)(14,21,25,28)(17,29,20,26)(18,27,34,30)(22,32,41,44)(23,39,36,37)(24,38,42,40)(31,33,35,43);(13,16,19,15)(14,22,25,41)(17,29,20,26)(18,33,34,43)(21,32,28,44)(23,40,36,38)(24,37,42,39)(27,35,30,31);(5,7,8,11)(6,10,9,12)(13,17,19,20)(14,23,25,36)(15,26,16,29)(18,31,34,35)(21,37,28,39)(22,40,41,38)(24,32,42,44)(27,43,30,33)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,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, 402170);