#6485469

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
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	#24662	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1510 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)(9,10)(11,15)(12,16)(13,14)(17,18)(19,29)(20,25)(21,26)(22,36)(23,27)(24,32)(28,37)(30,43)(31,44)(33,45)(34,42)(35,41)(38,46)(39,47)(40,48);(9,10)(11,16)(12,15)(13,14)(17,19,22,32,23,33,37,48)(18,24,28,29,27,40,36,45)(20,31,34,47,21,30,35,46)(25,39,41,44,26,38,42,43);(2,5)(6,8)(10,14)(15,16)(17,20)(18,25)(19,30)(21,23)(22,34)(24,38)(26,27)(28,41)(29,43)(31,33)(32,46)(35,37)(36,42)(39,40)(44,45)(47,48);(9,11,13,12)(10,15,14,16)(17,21,23,20)(18,26,27,25)(19,31,33,30)(22,35,37,34)(24,39,40,38)(28,42,36,41)(29,44,45,43)(32,47,48,46);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(18,27)(20,21)(24,40)(28,36)(29,45)(30,31)(34,35)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6392665);