#6637507

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636940
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	#37455	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1608 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,22,37)(20,30,33,46)(21,26,35,42)(25,28,40,32)(29,38,36,48)(31,45,34,47)(39,43,41,44);(2,5)(3,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,22)(18,24,27,37)(20,28,33,32)(21,29,35,36)(25,30,40,46)(26,38,42,48)(31,43,34,44)(39,45,41,47);(11,15)(13,16)(17,20,22,32,23,33,19,28)(18,25,24,30,27,40,37,46)(21,34,36,43,35,31,29,44)(26,41,38,47,42,39,48,45);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(17,21)(18,26)(19,29)(20,31)(22,36)(23,35)(24,38)(25,39)(27,42)(28,43)(30,45)(32,44)(33,34)(37,48)(40,41)(46,47);(2,5)(6,7)(10,14)(13,16)(21,35)(26,42)(29,36)(31,34)(38,48)(39,41)(43,44)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,d^c=di,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, 6544703);