#6356857

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356733
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	#61997	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1541 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)(3,10,13,16)(5,9,6,15)(8,11,14,12)(17,18,23,28)(19,29,33,39)(20,34,36,41)(21,26,22,27)(24,31,40,32)(25,37,42,35)(30,44,47,38)(43,45,48,46);(1,3,6,12,7,13,5,11)(2,8,9,10,4,14,15,16)(17,19,22,31,23,33,21,32)(18,24,26,39,28,40,27,29)(20,30,37,45,36,47,35,46)(25,38,41,48,42,44,34,43);(17,20,23,36)(18,25,28,42)(19,30,33,47)(21,35,22,37)(24,38,40,44)(26,41,27,34)(29,43,39,48)(31,45,32,46);(1,4,7,2)(3,8,13,14)(5,9,6,15)(10,11,16,12)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,36,37)(24,29,40,39)(25,41,42,34)(30,45,47,46)(38,43,44,48);(1,5,7,6)(2,9,4,15)(3,11,13,12)(8,10,14,16)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264053);