#439575

order	512
nilpotency class	3
isoclinism	isoclinic class of #422715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=efg,c^2=ef,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 346771);