#353831

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1722 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=f,g^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303331, #72303580, #72303585
maximal normal subgroup of	#72303331, #72303580, #72303585

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261027);