#6570606

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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	C₂×#496	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2061 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:

(2,4)(5,6,11,10)(7,8,9,12)(13,14)(15,24)(16,21)(17,22)(18,33)(19,23)(20,27)(25,41)(26,36)(28,35)(29,44)(30,38)(31,39)(32,40)(34,42)(37,43);(5,7,11,9)(6,12,10,8)(13,15,17,26,19,28,18,27)(14,20,33,35,23,36,22,24)(16,29,31,42,32,25,30,43)(21,37,38,41,40,34,39,44);(1,2,3,4)(5,8,11,12)(6,7,10,9)(13,16)(14,21)(15,25)(17,31)(18,30)(19,32)(20,34)(22,39)(23,40)(24,41)(26,43)(27,42)(28,29)(33,38)(35,44)(36,37);(5,9,11,7)(6,12,10,8)(13,17,19,18)(14,22,23,33)(15,26,28,27)(16,30,32,31)(20,24,36,35)(21,38,40,39)(25,42,29,43)(34,44,37,41);(5,10,11,6)(7,8,9,12)(13,17,19,18)(14,22,23,33)(15,27,28,26)(16,31,32,30)(20,35,36,24)(21,39,40,38)(25,42,29,43)(34,44,37,41)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=g,d^2=h,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,d^c=di,e^d=ehi

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, 6477802);