#6558116

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3269	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2485 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:

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

Polycyclic presentation:

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

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, 6465312);