#7144004

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#42880	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1635 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)(11,15)(14,16)(17,18,22,24)(19,28,31,40)(20,26,32,38)(21,27,23,35)(25,36,37,30)(29,43,44,48)(33,41,34,42)(39,45,47,46);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(18,24)(20,32)(25,37)(27,35)(28,40)(29,44)(33,34)(45,46);(9,10)(11,14)(12,13)(15,16)(17,19,22,31)(18,25,24,37)(20,29,32,44)(21,30,23,36)(26,39,38,47)(27,40,35,28)(33,45,34,46)(41,43,42,48);(3,8)(5,7)(10,13)(14,16)(17,20)(18,26)(19,29)(21,34)(22,32)(23,33)(24,38)(25,39)(27,42)(28,43)(30,46)(31,44)(35,41)(36,45)(37,47)(40,48);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,27,24,35)(19,30,31,36)(20,33,32,34)(25,40,37,28)(26,41,38,42)(29,45,44,46)(39,43,47,48)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,e^2=hi,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, 7051200);