#6468336

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1428 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)(9,10,13,16)(11,14,12,15)(17,18,22,26)(19,27,28,42)(20,24,32,40)(21,34,35,25)(23,36,37,30)(29,43,31,46)(33,48,44,41)(38,47,39,45);(11,12)(14,15)(17,19,21,30,22,28,35,36)(18,23,25,27,26,37,34,42)(20,31,33,47,32,29,44,45)(24,39,41,46,40,38,48,43);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(19,28)(20,32)(23,37)(24,40)(27,42)(30,36)(33,44)(41,48);(9,12,13,11)(10,15,16,14)(17,20)(18,24)(19,29)(21,33)(22,32)(23,38)(25,41)(26,40)(27,43)(28,31)(30,45)(34,48)(35,44)(36,47)(37,39)(42,46);(1,3,4,7)(2,6,5,8)(20,32)(24,40)(29,31)(33,44)(38,39)(41,48)(43,46)(45,47)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=hi,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,d^c=di,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, 6375532);