#6435863

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433049
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1582 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)(19,29,31,40)(20,34,35,42)(21,27,22,26)(24,33,39,32)(25,36,41,37)(30,38,44,48)(43,45,47,46);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,33,23,31,21,32)(18,24,27,40,28,39,26,29)(20,30,37,46,35,44,36,45)(25,38,34,47,41,48,42,43);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,21,36,23,35,22,37)(18,25,26,42,28,41,27,34)(19,30,32,45,31,44,33,46)(24,38,29,43,39,48,40,47);(9,11,12,15)(10,14,13,16)(20,35)(25,41)(30,44)(34,42)(36,37)(38,48)(43,47)(45,46);(19,31)(24,39)(29,40)(30,44)(32,33)(38,48)(43,47)(45,46)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343059);