#7017429

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

1607 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)(17,18,21,26,22,27,36,42)(19,28,30,45,31,23,37,40)(20,32,35,41,34,25,33,24)(29,39,48,38,47,44,46,43);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,31)(18,23,27,28)(20,29,34,47)(21,30,36,37)(24,38,41,43)(25,39,32,44)(26,40,42,45)(33,46,35,48);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,24)(19,29)(21,33)(22,34)(23,38)(25,26)(27,41)(28,43)(30,46)(31,47)(32,42)(35,36)(37,48)(39,40)(44,45);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20)(18,25)(19,29)(21,35)(22,34)(23,39)(24,26)(27,32)(28,44)(30,48)(31,47)(33,36)(37,46)(38,40)(41,42)(43,45);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,21,22,36)(18,26,27,42)(19,30,31,37)(20,33,34,35)(23,40,28,45)(24,25,41,32)(29,46,47,48)(38,39,43,44)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=fi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,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, 6924625);