#6362545

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362510
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1596 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ehi,f^a=fh,g^a=gi,d^b=df,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6269741);