#181158

order	512
nilpotency class	3
isoclinism	isoclinic class of #181134
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1509 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=hi,f^2=h,b^a=be,e^a=eh,g^a=ghi,f^b=fi,d^c=dhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88354);