#7094675

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1263 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)(3,6,4,7)(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)(28,41,43,47)(32,46,45,39)(36,42,48,44);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,28)(21,34)(22,30)(24,36)(25,40)(26,37)(27,33)(29,35)(31,43)(32,42)(38,48)(39,47)(41,46)(44,45);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20,22,31)(18,24,26,38)(19,28,30,43)(21,32,35,45)(23,36,37,48)(25,39,33,46)(27,41,40,47)(29,44,34,42);(11,16)(13,15)(19,29)(20,31)(21,35)(23,27)(24,38)(25,33)(28,42)(30,34)(36,47)(37,40)(41,48)(43,44);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(20,31)(24,38)(28,43)(32,45)(36,48)(39,46)(41,47)(42,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,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=eh,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, 7001871);