#6438458

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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2224 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)(5,6,8,9)(7,10,11,12)(13,14)(15,20)(16,21)(17,19)(18,22)(23,36)(24,35)(25,31)(26,38)(27,32)(28,33)(29,34)(30,37)(39,44)(40,42)(41,43);(6,9)(10,12)(14,19)(15,23)(16,26)(18,28)(20,34)(21,37)(25,29)(27,30)(31,36)(32,38)(35,42)(39,41);(2,4)(5,7,8,11)(6,10,9,12)(13,15,17,25)(14,20,19,31)(16,24,27,40)(18,29,28,23)(21,35,32,42)(22,34,33,36)(26,39,30,41)(37,43,38,44);(13,16,17,27)(14,21,19,32)(15,24,25,40)(18,30,28,26)(20,35,31,42)(22,37,33,38)(23,39,29,41)(34,43,36,44);(15,25)(20,31)(23,29)(24,40)(34,36)(35,42)(39,41)(43,44)

Polycyclic presentation:

2^9|a^2=fi,c^2=f,d^2=i,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, 6345654);