#7074650

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1520 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fh,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6981846);