#7101129

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1417 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=gi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,f^b=fi,d^c=dhi,e^c=ei,f^d=fi

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, 7008325);