#6357018

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356901
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#61986	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1530 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,d^2=hi,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ehi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264214);