#95199

order	512
nilpotency class	3
isoclinism	isoclinic class of #95196
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

1032 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gh,d^2=gi,b^a=bd,d^a=dg,f^a=fghi,c^b=cg,d^b=dh,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264585, #72264889
maximal normal subgroup of	#72264585, #72264889

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2395);