#180953

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1629 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88149);