#6357914

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357891
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1861 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,15)(14,16)(17,18,22,24)(19,27,30,40)(20,25,32,38)(21,34,28,26)(23,35,36,31)(29,41,44,48)(33,45,42,39)(37,46,47,43);(2,6)(7,14)(8,13)(9,15)(17,19,28,35,22,30,21,31)(18,23,34,40,24,36,26,27)(20,29,42,46,32,44,33,43)(25,37,45,48,38,47,39,41);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(18,24)(23,36)(25,38)(26,34)(27,40)(37,47)(39,45)(41,48);(3,10)(7,14)(9,15)(11,16)(17,20)(18,25)(19,29)(21,33)(22,32)(23,37)(24,38)(26,39)(27,41)(28,42)(30,44)(31,43)(34,45)(35,46)(36,47)(40,48);(19,30)(20,32)(23,36)(25,38)(27,40)(31,35)(33,42)(39,45)

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=gh,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=eh,d^c=di,e^d=eh

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, 6265110);