#7060454

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

1486 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)(17,18,21,26)(19,28,29,23)(20,25,33,41)(22,24,35,27)(30,40,32,37)(31,36,43,38)(34,39,46,42)(44,47,45,48);(1,2)(3,8)(4,7)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,32)(21,29)(22,31)(24,38)(25,40)(26,28)(27,36)(30,33)(34,45)(35,43)(37,41)(39,47)(42,48)(44,46);(9,11,12,16)(10,13,14,15)(18,24)(19,29)(22,35)(23,36)(25,39)(26,27)(28,38)(30,32)(34,46)(37,47)(40,48)(41,42);(1,3,5,4)(2,7,6,8)(11,16)(13,15)(17,20)(18,25)(19,30)(21,33)(22,34)(23,37)(24,39)(26,41)(27,42)(28,40)(29,32)(31,44)(35,46)(36,47)(38,48)(43,45);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(18,26)(20,33)(23,28)(24,27)(30,32)(34,46)(36,38)(44,45)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,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, 6967650);