#6558142

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2543 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,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, 6465338);