#7188549

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1251 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)(3,6)(4,8)(5,7)(10,14)(12,15)(17,18,22,25)(19,27,31,40)(20,32,33,24)(21,35,36,26)(23,29,38,30)(28,42,44,41)(34,39,48,47)(37,45,46,43);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,12,16,15)(17,19,22,31)(18,23,25,38)(20,28,33,44)(21,29,36,30)(24,37,32,46)(26,40,35,27)(34,43,48,45)(39,42,47,41);(2,7)(6,8)(9,11)(10,15)(12,14)(13,16)(17,20,22,33)(18,24,25,32)(19,28,31,44)(21,34,36,48)(23,37,38,46)(26,39,35,47)(27,41,40,42)(29,43,30,45);(1,4,5,3)(2,8,7,6)(10,14)(11,16)(18,25)(19,29)(21,36)(23,27)(24,32)(28,43)(30,31)(34,48)(37,41)(38,40)(42,46)(44,45);(9,12,13,15)(10,11,14,16)(18,25)(20,33)(23,38)(26,35)(27,40)(28,44)(34,48)(43,45)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=gi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi

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, 7095745);