#6358060

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358057
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

1820 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=g,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,e^b=ei,d^c=dhi,e^d=eh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265256);