#494911

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

884 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320128
maximal normal subgroup of	#72320128
maximal quotient of	#72320165

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402107);