#6538841

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1716 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6446037);