#94971

order	512
nilpotency class	3
isoclinism	isoclinic class of #94966
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gi,d^2=gh,e^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,e^b=eh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2167);