#495409

order	512
nilpotency class	4
isoclinism	isoclinic class of #480454
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×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1004 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)(3,7)(4,5)(6,8)(9,10,14,18,13,16,20,30)(11,19,23,34,22,15,27,29)(12,24,26,37,25,17,31,32)(21,28,36,39,35,33,38,40);(2,5)(6,7)(9,11,20,23)(10,15,18,29)(12,21,31,36)(13,22,14,27)(16,19,30,34)(17,28,32,39)(24,33,37,40)(25,35,26,38);(1,3,4,8)(2,6,5,7)(10,16)(15,19)(17,24)(18,30)(28,33)(29,34)(32,37)(39,40);(9,12,13,25)(10,17,16,24)(11,21,22,35)(14,26,20,31)(15,28,19,33)(18,32,30,37)(23,36,27,38)(29,39,34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=fi,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,g^b=gh,h^b=hi,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402605);