#6800675

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1475 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(17,18,22,26)(19,23,28,37)(20,25,32,42)(21,24,34,27)(29,36,31,40)(30,38,43,35)(33,39,46,41)(44,48,45,47);(2,5)(3,7)(9,10,12,13)(11,15,16,14)(17,19)(18,23)(20,31)(21,30)(22,28)(24,38)(25,40)(26,37)(27,35)(29,32)(33,45)(34,43)(36,42)(39,47)(41,48)(44,46);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(18,24)(19,28)(21,34)(23,35)(25,39)(26,27)(29,31)(33,46)(36,47)(37,38)(40,48)(41,42);(2,5)(6,8)(17,20,22,32)(18,25,26,42)(19,29,28,31)(21,33,34,46)(23,36,37,40)(24,39,27,41)(30,44,43,45)(35,47,38,48);(9,11,12,16)(10,14,13,15)(18,26)(19,28)(20,32)(24,27)(30,43)(33,46)(36,40)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=f,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6707871);