#7170834

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1353 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)(6,8)(10,13)(11,15)(17,18,22,27,21,26,35,24)(19,28,31,39,29,23,36,37)(20,25,34,42,33,43,46,40)(30,41,44,48,32,38,45,47);(1,2)(3,6)(4,5)(7,8)(10,13)(11,15)(17,19,21,29)(18,23,26,28)(20,32,33,30)(22,31,35,36)(24,39,27,37)(25,41,43,38)(34,45,46,44)(40,47,42,48);(1,3,4,7)(2,6,5,8)(9,10)(11,14)(12,13)(15,16)(18,24)(19,29)(20,33)(22,35)(23,37)(25,42)(26,27)(28,39)(38,48)(40,43)(41,47)(44,45);(2,5)(6,8)(10,13)(14,16)(17,20,21,33)(18,25,26,43)(19,30,29,32)(22,34,35,46)(23,38,28,41)(24,40,27,42)(31,44,36,45)(37,47,39,48);(9,11,12,15)(10,14,13,16)(18,26)(19,29)(20,33)(24,27)(31,36)(34,46)(38,41)(47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=fi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,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, 7078030);