#6971497

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	#37682	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1763 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)(3,6)(4,8)(5,7)(10,13)(11,15)(17,18)(19,27)(20,31)(21,23)(22,36)(24,30)(25,33)(26,35)(28,42)(29,44)(32,41)(34,40)(37,43)(38,48)(39,46)(45,47);(1,2,6,3)(4,7,5,8)(9,10,12,13)(11,14,15,16)(18,23)(19,28)(25,31)(26,36)(29,32)(30,43)(40,47)(46,48);(1,3,6,2)(4,7,5,8)(10,13)(14,16)(17,19,22,30,21,28,35,43)(18,24,26,27,23,37,36,42)(20,32,34,48,33,29,45,46)(25,39,40,44,31,38,47,41);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(17,20,21,33)(18,25,23,31)(19,29,28,32)(22,34,35,45)(24,38,37,39)(26,40,36,47)(27,41,42,44)(30,46,43,48);(1,5,6,4)(2,7,3,8)(20,33)(25,31)(29,32)(34,45)(38,39)(40,47)(41,44)(46,48)

Polycyclic presentation:

2^9|b^2=fi,c^2=g,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfhi,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, 6878693);