#6667560

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2780 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)(5,6)(7,11)(8,10)(9,12);(5,7)(6,9)(8,12)(10,11)(13,14)(15,24)(16,20)(17,21)(18,23)(19,26)(22,32)(25,40)(27,39)(28,42)(29,36)(30,38)(31,34)(33,41)(35,43)(37,44);(2,4)(13,15,18,27)(14,19,23,34)(16,28,30,25)(17,31,32,26)(20,35,38,33)(21,24,22,39)(29,41,44,43)(36,42,37,40);(6,10)(9,11)(13,16,18,30)(14,20,23,38)(15,25,27,28)(17,29,32,44)(19,33,34,35)(21,36,22,37)(24,40,39,42)(26,41,31,43);(7,12)(9,11)(14,21)(17,32)(19,24)(20,36)(22,23)(26,31)(29,44)(33,40)(34,39)(35,42)(37,38)(41,43)

Polycyclic presentation:

2^9|c^2=i,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6574756);