#6457449

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1640 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)(20,31)(21,24)(22,26)(23,37)(25,35)(27,32)(29,43)(30,39)(33,45)(34,46)(36,42)(38,48)(40,47)(41,44);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,21)(18,24,26,28)(20,29,34,33)(23,30,32,36)(25,38,40,41)(27,39,37,42)(31,45,46,43)(35,44,47,48);(11,15)(13,16)(17,20,32,47,22,34,23,35)(18,25,37,46,26,40,27,31)(19,29,36,48,21,33,30,44)(24,38,42,43,28,41,39,45);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,21,22,19)(18,24,26,28)(20,33,34,29)(23,36,32,30)(25,38,40,41)(27,39,37,42)(31,45,46,43)(35,48,47,44);(19,21)(24,28)(29,33)(30,36)(38,41)(39,42)(43,45)(44,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6364645);