#160434

order	512
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 67630);