#7047415

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1325 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)(10,14)(12,15)(17,18,22,24)(19,29,30,41)(20,26,34,39)(21,28,23,27)(25,32,38,33)(31,45,35,48)(36,44,37,43)(40,47,42,46);(1,3,5,8)(2,6,4,7)(9,10,13,14)(11,12,16,15)(18,24)(19,30)(20,34)(27,28)(32,33)(36,37)(40,42)(45,48);(9,11)(10,15)(12,14)(13,16)(17,19,21,32,22,30,23,33)(18,25,27,41,24,38,28,29)(20,35,37,46,34,31,36,47)(26,42,44,45,39,40,43,48);(3,8)(6,7)(10,14)(11,16)(17,20)(18,26)(19,31)(21,37)(22,34)(23,36)(24,39)(25,40)(27,44)(28,43)(29,45)(30,35)(32,47)(33,46)(38,42)(41,48);(1,4,5,2)(3,7,8,6)(9,12,13,15)(10,11,14,16)(17,21,22,23)(18,27,24,28)(19,32,30,33)(20,36,34,37)(25,41,38,29)(26,43,39,44)(31,46,35,47)(40,45,42,48)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=gi,e^2=f,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,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6954611);