#6490886

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469983
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×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×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3203 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)(7,11)(10,12)(13,14)(15,23)(16,20)(17,30)(18,21)(19,26)(22,32)(24,39)(25,38)(27,41)(28,36)(29,44)(31,34)(33,40)(35,42)(37,43);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,27)(17,31)(18,25)(20,35)(21,34)(22,38)(23,30)(24,28)(26,32)(29,40)(33,36)(37,39)(41,44)(42,43);(2,4)(6,9)(10,12)(13,16)(14,20)(15,24)(17,29)(18,28)(19,33)(21,36)(22,37)(23,39)(25,27)(26,40)(30,44)(31,42)(32,43)(34,35)(38,41);(6,9)(10,12)(15,25)(16,28)(19,34)(20,36)(23,38)(26,31)(29,43)(37,44);(5,7,8,11)(6,10,9,12)(14,21)(16,28)(19,34)(22,30)(23,38)(24,27)(29,43)(40,42)

Polycyclic presentation:

2^9|e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=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, 6398082);