#495386

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

850 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)(3,7)(4,5)(6,8)(10,15,19,23)(11,20)(12,16)(13,25)(17,31,32,39)(18,28,34,27)(21,35)(22,36)(24,30)(26,38,37,33);(2,5)(6,7)(9,10,16,28,25,34,12,23,14,19,30,27,13,18,24,15)(11,17,29,39,35,38,22,37,21,32,40,31,20,33,36,26);(1,3,4,8)(2,6,5,7)(9,11,13,20,14,21,25,35)(10,17,18,33,19,32,34,38)(12,22,16,29,24,36,30,40)(15,26,27,31,23,37,28,39);(11,21)(17,32)(20,35)(22,36)(26,37)(29,40)(31,39)(33,38)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,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	#72264964, #72264975, #72265035, …
maximal normal subgroup of	#72264964, #72264975, #72265035, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402582);