#6493161

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1457 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)(11,15)(14,16)(17,18)(19,30)(20,35)(21,27)(22,28)(23,29)(24,39)(25,34)(26,36)(31,41)(32,46)(33,43)(37,44)(38,42)(40,47)(45,48);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,31)(21,32)(22,38)(23,34)(24,33)(26,40)(27,41)(28,43)(29,30)(35,46)(36,48)(37,47)(39,42)(44,45);(10,13)(14,16)(17,20,23,21)(18,26,29,44)(19,31,34,32)(22,36,24,37)(25,40,30,45)(27,39,35,28)(33,47,38,48)(41,42,46,43);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,21,23,20)(18,27,29,35)(19,32,34,31)(22,36,24,37)(25,41,30,46)(26,39,44,28)(33,47,38,48)(40,42,45,43);(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,28,29,39)(19,33,34,38)(20,36,21,37)(25,42,30,43)(26,27,44,35)(31,47,32,48)(40,46,45,41)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=f,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, 6400357);