#7101212

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	#31068	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1431 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)(17,18,24,29)(19,30,20,26)(21,27,37,46)(22,28,23,38)(25,34,41,32)(31,43,33,42)(35,40,47,48)(36,44,39,45);(9,10)(11,15)(12,14)(13,16)(17,19,24,20)(18,25,29,41)(21,35,37,47)(22,32,23,34)(26,38,30,28)(27,43,46,42)(31,36,33,39)(40,45,48,44);(1,3,5,4)(2,6,8,7)(9,11,12,16)(10,13,14,15)(17,20,24,19)(18,26,29,30)(21,33,37,31)(22,32,23,34)(25,38,41,28)(27,42,46,43)(35,39,47,36)(40,45,48,44);(1,3,5,4)(2,7,8,6)(11,16)(13,15)(17,21)(18,27)(19,31)(20,33)(22,36)(23,39)(24,37)(25,40)(26,42)(28,44)(29,46)(30,43)(32,35)(34,47)(38,45)(41,48);(1,4,5,3)(2,6,8,7)(10,14)(13,15)(17,22,24,23)(18,28,29,38)(19,32,20,34)(21,36,37,39)(25,26,41,30)(27,44,46,45)(31,35,33,47)(40,42,48,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=ghi,d^2=g,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,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, 7008408);