#6461171

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1593 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)(11,13)(12,14)(15,16)(17,18,22,26)(19,28,21,24)(20,31,32,47)(23,37,36,27)(25,40,42,33)(29,44,34,46)(30,39,35,43)(38,45,41,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,29)(21,22)(23,30)(25,38)(26,28)(27,39)(31,46)(32,34)(33,45)(35,36)(37,43)(40,48)(41,42)(44,47);(10,14)(11,15)(17,20,23,33,22,32,36,40)(18,25,27,31,26,42,37,47)(19,29,30,45,21,34,35,48)(24,38,39,46,28,41,43,44);(11,15)(13,16)(17,21,22,19)(18,24,26,28)(20,29,32,34)(23,35,36,30)(25,41,42,38)(27,39,37,43)(31,44,47,46)(33,45,40,48);(1,3,4,7)(2,6,5,8)(19,21)(20,32)(24,28)(25,42)(30,35)(31,47)(33,40)(39,43)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6368367);