#6671483

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	#42880	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2161 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)(13,14,19,18)(15,20,25,32)(16,21,27,33)(17,29,31,22)(23,34,37,26)(24,30,39,35)(28,42,41,36)(38,40,44,43);(5,6,8,9)(7,10,11,12)(13,15,19,25)(14,20,18,32)(16,26,27,34)(17,30,31,35)(21,23,33,37)(22,24,29,39)(28,38,41,44)(36,43,42,40);(2,4)(13,16,19,27)(14,21,18,33)(15,23,25,37)(17,28,31,41)(20,34,32,26)(22,36,29,42)(24,38,39,44)(30,40,35,43);(5,7,8,11)(6,10,9,12)(16,27)(21,33)(23,37)(26,34)(28,41)(36,42)(38,44)(40,43);(6,9)(10,12)(13,17,19,31)(14,22,18,29)(15,24,25,39)(16,28,27,41)(20,35,32,30)(21,36,33,42)(23,38,37,44)(26,40,34,43)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6578679);