#6495982

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470428
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×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1504 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)(3,8)(4,6)(5,7)(11,15)(14,16)(17,18)(19,27)(20,24)(21,34)(22,25)(23,30)(26,35)(28,41)(29,42)(31,46)(32,39)(33,48)(36,43)(37,45)(38,47)(40,44);(9,10)(11,14)(12,13)(15,16)(17,19,21,30,22,28,35,43)(18,23,26,27,25,36,34,41)(20,31,33,47,32,29,44,45)(24,38,40,46,39,37,48,42);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(19,28)(20,32)(23,36)(24,39)(27,41)(30,43)(33,44)(40,48);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20)(18,24)(19,29)(21,33)(22,32)(23,37)(25,39)(26,40)(27,42)(28,31)(30,45)(34,48)(35,44)(36,38)(41,46)(43,47);(9,11,12,15)(10,14,13,16)(18,25)(19,28)(20,32)(26,34)(30,43)(33,44)(37,38)(42,46)

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=g,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di,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, 6403178);