#6470866

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
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 ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2130 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)(7,9)(8,10)(11,12)(13,14)(15,25)(16,30)(17,22)(18,23)(19,24)(20,27)(21,32)(26,34)(28,36)(29,35)(31,38)(33,37)(39,41)(40,43)(42,44);(5,7)(6,9)(8,11)(10,12)(13,15,18,28,19,29,17,27)(14,20,22,35,24,36,23,25)(16,26,32,41,33,42,31,40)(21,34,30,43,38,44,37,39);(2,4)(7,11)(9,12)(13,16,17,31,19,33,18,32)(14,21,23,37,24,38,22,30)(15,26,27,40,29,42,28,41)(20,34,25,39,36,44,35,43);(7,11)(9,12)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,31,33,32)(20,35,36,25)(21,30,38,37)(26,40,42,41)(34,43,44,39);(6,10)(9,12)(13,17,19,18)(14,22,24,23)(15,28,29,27)(16,31,33,32)(20,25,36,35)(21,30,38,37)(26,41,42,40)(34,39,44,43)

Polycyclic presentation:

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

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, 6378062);