#6438235

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2782 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)(7,10)(8,9)(11,12)(13,14,19,18)(15,20,25,31)(16,21,28,32)(17,22,30,33)(23,34,38,26)(24,35,39,29)(27,36,41,42)(37,43,44,40);(6,9)(10,12)(13,15,19,25)(14,20,18,31)(16,26,28,34)(17,29,30,35)(21,23,32,38)(22,24,33,39)(27,37,41,44)(36,43,42,40);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,23)(17,27)(18,32)(19,28)(20,34)(22,36)(24,37)(25,38)(26,31)(29,40)(30,41)(33,42)(35,43)(39,44);(13,17,19,30)(14,22,18,33)(15,24,25,39)(16,27,28,41)(20,35,31,29)(21,36,32,42)(23,37,38,44)(26,40,34,43);(16,28)(21,32)(23,38)(26,34)(27,41)(36,42)(37,44)(40,43)

Polycyclic presentation:

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

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, 6345431);