#7034350

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#42926	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1584 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)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,16,15)(17,18)(19,25)(20,35)(21,27)(22,28)(23,38)(24,29)(26,36)(30,44)(31,40)(32,41)(33,47)(34,42)(37,43)(39,45)(46,48);(2,5)(3,7)(9,10)(11,15)(12,13)(14,16)(17,19,24,34)(18,25,29,42)(20,30,21,31)(22,32,23,33)(26,39,43,48)(27,40,35,44)(28,41,38,47)(36,45,37,46);(10,13)(11,16)(17,20,24,21)(18,26,29,43)(19,30,34,31)(22,36,23,37)(25,39,42,48)(27,38,35,28)(32,45,33,46)(40,47,44,41);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,14,13,15)(17,21,24,20)(18,27,29,35)(19,31,34,30)(22,36,23,37)(25,40,42,44)(26,38,43,28)(32,45,33,46)(39,47,48,41);(10,13)(14,15)(17,22,24,23)(18,28,29,38)(19,32,34,33)(20,36,21,37)(25,41,42,47)(26,27,43,35)(30,45,31,46)(39,40,48,44)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6941546);