#6434252

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1783 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:

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

Polycyclic presentation:

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

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, 6341448);