#6432292

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431707
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2264 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ei,h^b=hi,h^c=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, 6339488);