#182035

order	512
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂²⋊C₄
commutator	C₂²	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1344 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=i,f^2=h,g^2=i,b^a=be,g^a=gh,e^b=eh

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89231);