#6358064

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,d^2=h,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, 6265260);