#7016506

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1327 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=f,d^2=fi,e^2=fi,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, 6923702);