#361110

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

774 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)(3,8)(4,6)(5,7)(9,10,14,20,15,21,34,19)(11,22,25,16,27,36,26,37)(12,28,24,17,30,39,29,35)(13,31,32,18,33,38,23,40);(1,3,5,4)(2,6,7,8)(9,11,14,25,15,27,34,26)(10,16,19,22,21,37,20,36)(12,13,24,32,30,33,29,23)(17,38,28,18,35,31,39,40);(1,4,5,3)(2,6,7,8)(9,12,14,24,15,30,34,29)(10,17,19,28,21,35,20,39)(11,23,25,13,27,32,26,33)(16,18,22,31,37,40,36,38);(2,7)(6,8)(9,13,14,32,15,33,34,23)(10,18,19,31,21,40,20,38)(11,24,25,30,27,29,26,12)(16,35,22,39,37,17,36,28)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268306);