#7170544

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	#42880	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1572 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)(10,13)(11,15)(17,18,21,26,22,23,35,38)(19,27,31,36,30,40,29,24)(20,25,34,42,32,37,33,41)(28,43,45,47,44,48,46,39);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(18,23)(20,32)(24,36)(26,38)(27,40)(28,44)(33,34)(45,46);(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,23,36)(20,28,32,44)(21,29,35,31)(25,39,37,47)(26,40,38,27)(33,45,34,46)(41,43,42,48);(3,8)(5,7)(10,13)(14,16)(17,20)(18,25)(19,28)(21,34)(22,32)(23,37)(24,39)(26,42)(27,43)(29,46)(30,44)(31,45)(33,35)(36,47)(38,41)(40,48);(9,11,12,15)(10,14,13,16)(17,21,22,35)(18,26,23,38)(19,29,30,31)(20,33,32,34)(24,40,36,27)(25,41,37,42)(28,45,44,46)(39,43,47,48)

Polycyclic presentation:

2^9|a^2=fghi,b^2=fi,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,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, 7077740);