#364694

order	512
nilpotency class	3
isoclinism	isoclinic class of #23876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1528 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)(9,10,14,18,15,19,13,26)(11,16,22,32,21,30,27,31)(12,17,24,35,25,36,28,34)(20,37,39,40,38,29,23,33);(2,5)(6,8)(9,11,14,22,15,21,13,27)(10,16,18,32,19,30,26,31)(12,23,24,20,25,39,28,38)(17,33,35,37,36,40,34,29);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,20)(13,28)(14,24)(15,25)(16,29)(18,35)(19,36)(21,38)(22,39)(23,27)(26,34)(30,37)(31,40)(32,33);(2,5)(6,8)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271890);