#6468683

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1547 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)(3,6)(4,5)(7,8)(10,14)(15,16)(17,18,24,38,23,29,22,28)(19,30,37,25,34,44,33,41)(20,26,21,27,35,42,36,43)(31,40,47,39,46,48,32,45);(9,10,13,14)(11,16,12,15)(17,19)(18,25)(20,31)(21,32)(22,37)(23,34)(24,33)(26,39)(27,40)(28,44)(29,41)(30,38)(35,46)(36,47)(42,45)(43,48);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,23,35)(18,26,29,42)(19,31,34,46)(21,22,36,24)(25,39,41,45)(27,28,43,38)(30,40,44,48)(32,37,47,33);(9,12,13,11)(10,16,14,15)(17,21)(18,27)(19,32)(20,22)(23,36)(24,35)(25,40)(26,28)(29,43)(30,45)(31,37)(33,46)(34,47)(38,42)(39,44)(41,48);(1,3,4,7)(2,6,5,8)(20,35)(21,36)(26,42)(27,43)(31,46)(32,47)(39,45)(40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375879);