#6495590

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1409 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)(17,18,22,25)(19,27,30,44)(20,24,32,41)(21,34,35,26)(23,36,39,31)(28,42,29,43)(33,48,45,40)(37,47,38,46);(9,10)(11,14)(12,13)(15,16)(17,19,21,31,22,30,35,36)(18,23,26,27,25,39,34,44)(20,29,33,47,32,28,45,46)(24,38,40,43,41,37,48,42);(2,5)(6,8)(10,13)(14,16)(17,20)(18,24)(19,28)(21,33)(22,32)(23,37)(25,41)(26,40)(27,42)(29,30)(31,46)(34,48)(35,45)(36,47)(38,39)(43,44);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20)(18,24)(19,29)(21,33)(22,32)(23,38)(25,41)(26,40)(27,43)(28,30)(31,47)(34,48)(35,45)(36,46)(37,39)(42,44);(9,11,12,15)(10,14,13,16)(18,25)(19,30)(20,32)(26,34)(31,36)(33,45)(37,38)(42,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,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=dhi,e^b=ei,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6402786);