#6454464

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1994 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361660);