#6516796

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500076
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

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

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, 6423992);