#6432416

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3250 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339612);