#6432326

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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	#23386	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2348 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)(7,10)(8,9)(11,12)(13,14,18,24,19,22,17,23)(15,20,28,35,26,36,27,34)(16,29,30,37,31,38,32,21)(25,39,41,43,40,44,42,33);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,25)(17,27)(18,28)(19,26)(21,33)(22,36)(23,34)(24,35)(29,39)(30,41)(31,40)(32,42)(37,43)(38,44);(2,4)(13,16)(14,21)(15,25)(17,30)(18,32)(19,31)(20,33)(22,37)(23,29)(24,38)(26,40)(27,41)(28,42)(34,39)(35,44)(36,43);(5,7,8,11)(6,10,9,12)(14,22)(15,26)(21,37)(23,24)(25,40)(27,28)(29,38)(41,42);(13,17,19,18)(14,23,22,24)(15,27,26,28)(16,30,31,32)(20,34,36,35)(21,29,37,38)(25,41,40,42)(33,39,43,44)

Polycyclic presentation:

2^9|a^2=g,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,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, 6339522);