#6357609

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357552
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

1821 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)(3,9)(4,12)(5,10)(6,15)(7,11)(8,13)(14,16)(17,18)(19,27)(20,25)(21,34)(22,24)(23,30)(26,28)(29,40)(31,39)(32,37)(33,45)(35,41)(36,44)(38,42)(43,47)(46,48);(1,3,5,4,7,14,6,13)(2,8,15,16,11,12,10,9)(17,19,28,41,22,31,21,30)(18,23,34,39,24,35,26,27)(20,29,42,48,32,43,33,44)(25,36,45,47,37,46,38,40);(1,4,6,3,7,13,5,14)(2,9,10,12,11,16,15,8)(18,24)(23,35)(25,37)(26,34)(27,39)(36,46)(38,45)(40,47);(1,2)(3,12)(4,9)(5,15)(6,10)(7,11)(8,14)(13,16)(17,20)(18,25)(19,29)(21,33)(22,32)(23,36)(24,37)(26,38)(27,40)(28,42)(30,44)(31,43)(34,45)(35,46)(39,47)(41,48);(1,5,7,6)(2,10,11,15)(3,4,14,13)(8,9,12,16)(20,32)(25,37)(29,43)(33,42)(36,46)(38,45)(40,47)(44,48)

Polycyclic presentation:

2^9|b^2=fh,c^2=ghi,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=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, 6264805);