#7101275

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	#31066	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1438 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)(17,18,22,26)(19,27,30,40)(20,24,31,38)(21,33,35,25)(23,34,37,29)(28,41,43,48)(32,45,44,39)(36,46,47,42);(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,26,37)(20,28,31,43)(21,34,35,29)(24,36,38,47)(25,40,33,27)(32,46,44,42)(39,48,45,41);(1,3,5,4)(2,6,8,7)(9,11,12,16)(10,13,14,15)(17,20,22,31)(18,24,26,38)(19,28,30,43)(21,32,35,44)(23,36,37,47)(25,39,33,45)(27,41,40,48)(29,42,34,46);(1,3,5,4)(2,7,8,6)(11,16)(13,15)(19,29)(21,35)(23,27)(25,33)(28,42)(30,34)(32,44)(36,41)(37,40)(39,45)(43,46)(47,48);(1,4,5,3)(2,6,8,7)(10,14)(13,15)(19,30)(20,31)(23,37)(24,38)(27,40)(29,34)(32,44)(39,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=ghi,d^2=g,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,d^c=dh,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, 7008471);