#6453810

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23923	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3157 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:

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

Polycyclic presentation:

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

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, 6361006);