#7007685

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1424 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,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, 6914881);