#495566

order	512
nilpotency class	4
isoclinism	isoclinic class of #481349
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60682	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

828 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)(3,6)(4,5)(7,8)(10,15,18,23)(12,16)(13,25)(17,26,31,36)(19,28,33,27)(21,29)(22,35)(24,30)(32,38,40,37)(34,39);(9,10,16,28,25,33,12,23,14,18,30,27,13,19,24,15)(11,17,29,38,35,40,21,36,20,31,39,37,22,32,34,26);(2,5)(6,8)(9,11)(10,17)(12,21)(13,22)(14,20)(15,26)(16,29)(18,31)(19,32)(23,36)(24,34)(25,35)(27,37)(28,38)(30,39)(33,40);(1,3,4,7)(2,6,5,8)(11,20)(17,31)(21,34)(22,35)(26,36)(29,39)(32,40)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313899, #72313905, #72313935, …
maximal normal subgroup of	#72313899, #72313905, #72313935, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402762);