#95322

order	512
nilpotency class	3
isoclinism	isoclinic class of #95196
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

768 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gh,d^2=gi,e^2=gh,f^2=gh,b^a=bd,c^a=ch,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2518);