#7060708

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

1531 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)(19,28)(20,25)(21,26)(22,24)(23,29)(27,35)(30,40)(31,36)(32,37)(33,42)(34,39)(38,43)(41,46)(44,47)(45,48);(1,3)(2,6)(4,7)(5,8)(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)(37,42)(39,47)(41,48)(44,46);(9,11,12,16)(10,13,14,15)(18,24)(19,29)(20,33)(22,35)(23,36)(25,41)(26,27)(28,38)(37,48)(39,42)(40,47)(44,45);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20,21,33)(18,25,26,42)(19,30,29,32)(22,34,35,46)(23,37,28,40)(24,39,27,41)(31,44,43,45)(36,47,38,48);(1,4,5,2)(3,7,8,6)(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=fi,c^2=hi,d^2=f,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=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, 6967904);