#6607199

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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

2331 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)(5,6)(7,10)(8,11)(9,12)(13,14)(15,25)(16,30)(17,22)(18,23)(19,24)(20,29)(21,33)(26,35)(27,36)(28,37)(31,38)(32,40)(34,39)(41,42)(43,44);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,29,19,28,18,27)(14,20,23,25,24,36,22,37)(16,31,32,42,34,26,33,43)(21,38,39,44,40,35,30,41);(7,12)(9,10)(13,16,18,33,19,34,17,32)(14,21,22,30,24,40,23,39)(15,26,27,42,28,31,29,43)(20,35,37,44,36,38,25,41);(5,7)(6,10)(8,12)(9,11)(13,17,19,18)(14,22,24,23)(15,27,28,29)(16,32,34,33)(20,25,36,37)(21,30,40,39)(26,42,31,43)(35,41,38,44);(7,12)(9,10)(15,28)(20,36)(25,37)(26,31)(27,29)(35,38)(41,44)(42,43)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6514395);