#6357742

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357716
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
prime radical	#3511	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1775 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=gi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=di,e^c=eh,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, 6264938);