#6576871

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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
prime radical	#35687	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2670 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=ehi,h^d=hi,h^e=hi

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, 6484067);