#6629680

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628847
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2270 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)(13,14,18,21)(15,20,25,35)(16,29,31,38)(17,24,19,23)(22,32,37,30)(26,40,39,44)(27,36,28,34)(33,42,43,41);(1,2,4,3)(5,7,8,11)(6,9,10,12)(13,15)(14,20)(16,26)(17,27)(18,25)(19,28)(21,35)(22,33)(23,34)(24,36)(29,40)(30,41)(31,39)(32,42)(37,43)(38,44);(6,10)(9,12)(14,21)(15,25)(22,37)(23,24)(26,39)(27,28)(29,38)(41,42);(1,3,4,2)(13,16)(14,22)(15,26)(17,30)(18,31)(19,32)(20,33)(21,37)(23,29)(24,38)(25,39)(27,41)(28,42)(34,40)(35,43)(36,44);(13,17,18,19)(14,23,21,24)(15,27,25,28)(16,30,31,32)(20,34,35,36)(22,29,37,38)(26,41,39,42)(33,40,43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536876);