#6449068

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3372 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=gi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,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, 6356264);