#7170289

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

1422 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=i,c^2=i,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7077485);