#6483116

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
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 ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2060 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)(15,19)(16,26)(17,31)(18,21)(20,29)(22,28)(23,37)(24,33)(25,40)(27,41)(30,42)(32,39)(34,38)(35,43)(36,44);(2,4)(7,8)(10,12)(13,15)(14,19)(16,27)(17,25)(18,24)(20,35)(21,33)(22,34)(23,30)(26,41)(28,38)(29,43)(31,40)(32,36)(37,42)(39,44);(5,7,9,8)(6,10,11,12)(13,16,28,44,18,30,17,29)(14,20,31,42,21,36,22,26)(15,23,38,43,24,27,25,39)(19,32,40,41,33,35,34,37);(5,8,9,7)(6,10,11,12)(15,24)(19,33)(23,27)(25,38)(32,35)(34,40)(37,41)(39,43);(6,11)(10,12)(14,21)(15,24)(20,36)(22,31)(23,27)(25,38)(26,42)(39,43)

Polycyclic presentation:

2^9|c^2=gi,d^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,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, 6390312);