#6668859

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2052 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);(6,9)(7,11)(13,14,18,22)(15,23,24,37)(16,27,29,38)(17,30,33,21)(19,31,34,26)(20,32,36,28)(25,35,40,43)(39,42,44,41);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,25)(17,31)(18,24)(20,35)(21,37)(22,34)(23,30)(26,33)(27,39)(28,42)(29,40)(32,41)(36,43)(38,44);(6,9)(10,12)(15,24)(19,34)(23,37)(25,40)(26,31)(35,43)(39,44)(41,42);(13,16,18,29)(14,20,22,36)(15,25,24,40)(17,32,33,28)(19,35,34,43)(21,38,30,27)(23,39,37,44)(26,42,31,41)

Polycyclic presentation:

2^9|b^2=i,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6576055);