#7007358

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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1440 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)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,29,24,30,22,28)(19,25,33,41,34,42,38,44)(20,35,36,43,21,27,37,26)(31,45,47,48,32,40,46,39);(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,31)(21,32)(22,38)(23,33)(24,34)(26,39)(27,40)(28,44)(29,41)(30,42)(35,45)(36,47)(37,46)(43,48);(9,11)(10,13)(12,16)(14,15)(17,20,24,21)(18,26,30,43)(19,31,34,32)(22,36,23,37)(25,39,42,48)(27,28,35,29)(33,46,38,47)(40,44,45,41);(1,3,4,8)(2,6,5,7)(11,16)(13,15)(17,21,24,20)(18,27,30,35)(19,32,34,31)(22,36,23,37)(25,40,42,45)(26,28,43,29)(33,46,38,47)(39,44,48,41);(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,33,34,38)(20,36,21,37)(25,41,42,44)(26,35,43,27)(31,46,32,47)(39,40,48,45)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,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, 6914554);