#495525

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂×#3176	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

908 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,d^2=f,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,e^b=eh,g^b=gh,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, 402721);