#6357859

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357820
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

1845 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)(3,10)(4,9)(5,7)(6,14)(8,13)(11,16)(12,15)(17,18)(19,27)(20,25)(21,33)(22,24)(23,29)(26,34)(28,40)(30,39)(31,37)(32,45)(35,43)(36,42)(38,44)(41,47)(46,48);(2,7)(8,15)(9,14)(10,16)(17,19,21,29,22,30,34,43)(18,23,26,27,24,35,33,39)(20,28,32,42,31,41,44,48)(25,36,38,40,37,46,45,47);(1,3,6,13,5,11,4,12)(2,8,9,10,7,15,14,16)(18,24)(23,35)(25,37)(26,33)(27,39)(36,46)(38,45)(40,47);(3,11)(8,15)(10,16)(12,13)(17,20)(18,25)(19,28)(21,32)(22,31)(23,36)(24,37)(26,38)(27,40)(29,42)(30,41)(33,45)(34,44)(35,46)(39,47)(43,48);(1,4,5,6)(2,9,7,14)(3,12,11,13)(8,10,15,16)(20,31)(25,37)(28,41)(32,44)(36,46)(38,45)(40,47)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265055);