#6493909

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1510 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)(17,18,23,28)(19,29,32,25)(20,24,34,40)(21,26,36,43)(22,38,30,27)(31,45,35,41)(33,39,47,42)(37,48,46,44);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,30,34,23,32,22,20)(18,24,38,25,28,40,27,29)(21,35,46,33,36,31,37,47)(26,42,48,45,43,39,44,41);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,22,32,23,34,30,19)(18,25,27,24,28,29,38,40)(21,33,37,35,36,47,46,31)(26,41,44,39,43,45,48,42);(10,13)(14,16)(17,21)(18,26)(19,31)(20,33)(22,37)(23,36)(24,39)(25,41)(27,44)(28,43)(29,45)(30,46)(32,35)(34,47)(38,48)(40,42);(9,11,12,15)(10,14,13,16)(21,36)(26,43)(31,35)(33,47)(37,46)(39,42)(41,45)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=f,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,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, 6401105);