#6668898

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	C₂×#3283	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2082 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)(13,14,18,23)(15,25,28,20)(16,21,19,24)(17,30,32,22)(26,34,40,39)(27,38,33,35)(29,37,31,36)(41,43,42,44);(6,9)(7,11)(13,15,18,28)(14,20,23,25)(16,27,19,33)(17,26,32,40)(21,35,24,38)(22,34,30,39)(29,41,31,42)(36,43,37,44);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(17,31)(18,19)(22,37)(23,24)(26,40)(27,33)(29,32)(30,36)(34,39)(35,38);(6,9)(10,12)(13,17,18,32)(14,22,23,30)(15,26,28,40)(16,29,19,31)(20,34,25,39)(21,36,24,37)(27,41,33,42)(35,43,38,44);(15,27)(16,19)(20,35)(21,24)(25,38)(26,41)(28,33)(29,31)(34,43)(36,37)(39,44)(40,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^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,e^c=ei,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6576094);