#95145

order	512
nilpotency class	3
isoclinism	isoclinic class of #95118
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

850 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,5,7,8)(4,6)(9,10)(11,17,19,14)(12,20)(13,16)(15,21)(18,22,24,23)(25,26,28,32,29,31,37,39)(27,33,35,40,34,30,36,38);(1,3)(2,5,6,8)(4,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27)(26,30)(28,36)(29,34)(31,33)(32,40)(35,37)(38,39);(2,6)(5,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(26,31)(27,34)(32,39)(35,36)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ghi,e^2=i,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, 2341);