#6629684

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628851
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2289 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536880);