#6358038

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358021
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

1866 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=ei,d^c=dhi,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, 6265234);