#6930495

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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	#37379	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1533 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6837691);