#6516788

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500060
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

3386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6423984);