#181730

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1874 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,5,8,6,3,4,7)(9,10)(11,19)(12,17)(13,18)(14,16)(15,21)(20,23)(22,24);(1,3,6,2)(4,8,5,7)(9,11,13,22,14,20,12,21)(10,15,17,23,16,24,18,19);(2,3)(7,8)(10,16)(11,20)(17,18)(21,22);(1,4,6,5)(2,7,3,8)(9,12,14,13)(10,17,16,18)(11,21,20,22)(15,23,24,19)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=ch,e^a=eh,g^a=ghi,c^b=ch,e^b=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239188, #72239277, #72239287, …
maximal normal subgroup of	#72239188, #72239277, #72239287, …
maximal quotient of	#72239229, #72239594

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88926);