#6407557

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂⁴
commutator, socle	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#34547	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1613 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=cfgi,d^b=dfi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6314753);