#6471907

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470137
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	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1780 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)(17,18,22,27)(19,28,31,24)(20,33,35,40)(21,26,23,36)(25,37,41,34)(29,44,46,48)(30,43,32,39)(38,45,42,47);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,31)(18,24,27,28)(20,29,35,46)(21,30,23,32)(25,38,41,42)(26,39,36,43)(33,48,40,44)(34,45,37,47);(11,15)(13,16)(17,20,22,35)(18,25,27,41)(19,29,31,46)(21,34,23,37)(24,38,28,42)(26,40,36,33)(30,45,32,47)(39,44,43,48);(11,15)(13,16)(17,21,22,23)(18,26,27,36)(19,30,31,32)(20,34,35,37)(24,39,28,43)(25,40,41,33)(29,45,46,47)(38,44,42,48);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(19,31)(24,28)(29,46)(30,32)(38,42)(39,43)(44,48)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=fi,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

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, 6379103);