#6910259

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#39357	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1837 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6817455);