#7002915

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	#42796	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1531 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)(9,10)(11,13)(12,14)(15,16)(17,18,22,26,21,23,35,38)(19,27,30,37,28,43,36,24)(20,31,34,47,32,25,40,42)(29,45,46,39,33,44,48,41);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(18,23)(19,28)(20,32)(26,38)(30,36)(34,40)(39,41)(44,45);(9,11)(10,13)(12,16)(14,15)(17,19,21,28)(18,24,23,37)(20,33,32,29)(22,36,35,30)(25,41,31,39)(26,43,38,27)(34,46,40,48)(42,44,47,45);(3,8)(5,7)(11,16)(13,15)(17,20)(18,25)(19,29)(21,32)(22,34)(23,31)(24,39)(26,42)(27,44)(28,33)(30,46)(35,40)(36,48)(37,41)(38,47)(43,45);(10,14)(13,15)(18,23)(20,32)(24,37)(26,38)(27,43)(29,33)(34,40)(46,48)

Polycyclic presentation:

2^9|a^2=fgi,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6910111);