#6671527

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42648	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2149 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)(6,9)(7,11)(13,14,17,21)(15,23,25,19)(16,27,29,20)(18,22,31,37)(24,38,32,33)(26,34,41,39)(28,36,30,35)(40,43,42,44);(5,6,8,9)(7,10,11,12)(15,24)(16,28)(18,31)(19,33)(20,35)(22,37)(23,38)(25,32)(26,41)(27,36)(29,30)(34,39);(2,4)(13,15,17,25)(14,19,21,23)(16,26,29,41)(18,32,31,24)(20,34,27,39)(22,38,37,33)(28,42,30,40)(35,44,36,43);(5,7,8,11)(6,10,9,12)(15,25)(19,23)(24,32)(26,41)(33,38)(34,39)(40,42)(43,44);(6,9)(10,12)(13,16,17,29)(14,20,21,27)(15,26,25,41)(18,28,31,30)(19,34,23,39)(22,35,37,36)(24,40,32,42)(33,43,38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,d^c=di,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, 6578723);