#6594127

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1416 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)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,33)(21,25)(22,27)(23,31)(24,35)(26,29)(30,44)(32,37)(34,40)(36,45)(38,42)(39,43)(41,46)(47,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,29,42,22,32,21,31)(18,23,25,37,27,38,26,28)(20,30,43,48,34,45,35,46)(24,36,40,47,39,44,33,41);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,21,35,22,34,29,43)(18,24,26,33,27,39,25,40)(19,30,31,46,32,45,42,48)(23,36,28,41,38,44,37,47);(9,11)(10,13)(12,15)(14,16)(20,34)(24,39)(30,45)(33,40)(35,43)(36,44)(41,47)(46,48);(11,15)(13,16)(17,21,22,29)(18,25,27,26)(19,31,32,42)(20,35,34,43)(23,37,38,28)(24,40,39,33)(30,46,45,48)(36,47,44,41)

Polycyclic presentation:

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

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, 6501323);