#6492963

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1394 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)(11,15)(14,16)(17,18)(19,25)(20,34)(21,27)(22,28)(23,38)(24,29)(26,35)(30,45)(31,40)(32,41)(33,43)(36,44)(37,42)(39,47)(46,48);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,30)(21,31)(22,37)(23,32)(24,33)(26,39)(27,40)(28,42)(29,43)(34,45)(35,47)(36,46)(38,41)(44,48);(10,13)(14,16)(17,20,24,21)(18,26,29,44)(19,30,33,31)(22,35,23,36)(25,39,43,48)(27,38,34,28)(32,46,37,47)(40,41,45,42);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,21,24,20)(18,27,29,34)(19,31,33,30)(22,35,23,36)(25,40,43,45)(26,38,44,28)(32,46,37,47)(39,41,48,42);(9,11,12,15)(10,14,13,16)(17,22,24,23)(18,28,29,38)(19,32,33,37)(20,35,21,36)(25,41,43,42)(26,27,44,34)(30,46,31,47)(39,45,48,40)

Polycyclic presentation:

2^9|c^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6400159);