#95332

order	512
nilpotency class	3
isoclinism	isoclinic class of #7035
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	C₂×#95
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

800 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 170 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)(3,7)(4,6)(5,8)(9,10,12,17,14,18,13,16)(11,19,20,15,22,23,21,24)(25,26)(27,33,35,30)(28,31)(29,32)(34,39,40,37)(36,38);(1,3)(2,5,6,7)(4,8)(9,11,13,21,14,22,12,20)(10,15,17,23,18,24,16,19)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239114, #72239116, #72239123
maximal normal subgroup of	#72239114, #72239116, #72239123

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2528);