#6536721

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1438 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)(3,7)(4,8)(5,6)(17,18,23,27)(19,28,32,43)(20,33,34,25)(21,26,22,37)(24,30,39,31)(29,44,45,42)(35,41,36,40)(38,46,48,47);(9,10)(11,14)(12,13)(15,16)(17,19,21,30,23,32,22,31)(18,24,37,43,27,39,26,28)(20,29,36,47,34,45,35,46)(25,38,40,44,33,48,41,42);(2,6)(7,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,36)(22,35)(23,34)(24,38)(26,41)(27,33)(28,42)(30,47)(31,46)(32,45)(37,40)(39,48)(43,44);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(20,34)(25,33)(29,45)(35,36)(38,48)(40,41)(42,44)(46,47);(1,4,5,3)(2,8,6,7)(11,15)(14,16)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,35,34,36)(24,28,39,43)(25,40,33,41)(29,46,45,47)(38,44,48,42)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=gi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^c=ei,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6443917);