#7072343

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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1375 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fh,e^2=f,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, 6979539);