#7010627

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1270 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)(9,10)(11,13)(12,14)(15,16)(17,18,23,27,22,28,21,37)(19,29,31,42,32,24,38,43)(20,33,35,45,36,26,39,25)(30,41,48,40,34,46,47,44);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,32)(18,24,28,29)(20,34,36,30)(21,38,23,31)(25,44,45,40)(26,46,33,41)(27,43,37,42)(35,47,39,48);(9,11)(10,13)(12,16)(14,15)(17,20,22,36)(18,25,28,45)(19,30,32,34)(21,35,23,39)(24,40,29,44)(26,37,33,27)(31,47,38,48)(41,42,46,43);(2,5)(6,8)(11,16)(13,15)(17,20)(18,26)(19,30)(21,39)(22,36)(23,35)(24,41)(25,27)(28,33)(29,46)(31,48)(32,34)(37,45)(38,47)(40,43)(42,44);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,22,23)(18,27,28,37)(19,31,32,38)(20,35,36,39)(24,42,29,43)(25,26,45,33)(30,47,34,48)(40,46,44,41)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=f,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, 6917823);