#6524316

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1482 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6431512);