#6633285

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632068
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	#23939	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1684 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:

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

Polycyclic presentation:

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

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, 6540481);