#6633322

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1615 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)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18,23,29)(19,27,21,25)(20,33,36,44)(22,30,24,28)(26,38,42,35)(31,45,34,48)(32,40,37,43)(39,47,41,46);(1,3)(2,5)(4,7)(6,8)(9,10,13,14)(11,16,12,15)(17,19,23,21)(18,25,29,27)(20,31,36,34)(22,37,24,32)(26,39,42,41)(28,43,30,40)(33,48,44,45)(35,47,38,46);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(17,20)(18,26)(19,31)(21,34)(22,35)(23,36)(24,38)(25,39)(27,41)(28,33)(29,42)(30,44)(32,46)(37,47)(40,45)(43,48);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(17,21,23,19)(18,27,29,25)(20,34,36,31)(22,32,24,37)(26,41,42,39)(28,40,30,43)(33,45,44,48)(35,46,38,47);(2,6)(5,8)(17,22,23,24)(18,28,29,30)(19,32,21,37)(20,35,36,38)(25,40,27,43)(26,33,42,44)(31,46,34,47)(39,45,41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6540518);