#95184

order	512
nilpotency class	3
isoclinism	isoclinic class of #95118
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

844 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)(3,7,8,5)(4,6)(9,10,12,15,13,14,11,16)(17,18,21,19,20,23,22,24);(1,3)(2,5)(4,8)(6,7)(10,14)(11,12)(17,19,22,23,20,24,21,18);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(18,23)(19,24)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=hi,e^2=i,f^2=h,b^a=bd,c^a=cg,d^a=dg,f^a=fh,c^b=ci,d^b=dh,e^b=eg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2380);