#495572

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	C₂×#3176	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

860 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)(9,10,14,18,15,20,29,19)(11,21,17,28,24,34,23,13)(12,16,27,32,25,30,39,33)(22,35,31,38,36,40,37,26);(9,11,13,10,14,23,21,19,15,24,28,20,29,17,34,18)(12,22,26,16,27,37,35,33,25,36,38,30,39,31,40,32);(2,5)(6,8)(9,12,15,25)(10,16,20,30)(11,22,24,36)(13,26,28,38)(14,27,29,39)(17,31,23,37)(18,32,19,33)(21,35,34,40);(1,3,4,7)(2,6,5,8)(12,25)(16,30)(22,36)(26,38)(27,39)(31,37)(32,33)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=i,d^2=f,e^2=h,g^2=i,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	#72313905, #72313923, #72313928, …
maximal normal subgroup of	#72313905, #72313923, #72313928, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402768);