#6989053

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1511 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,7)(3,8,4,6)(9,10,12,14)(11,15,16,13)(17,18,23,27,22,26,21,28)(19,29,38,41,31,24,32,40)(20,33,35,25,36,44,37,43)(30,45,46,42,34,48,47,39);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,26,29)(20,34,36,30)(21,32,23,38)(25,42,43,39)(27,40,28,41)(33,45,44,48)(35,47,37,46);(1,2)(3,8)(4,6)(5,7)(11,16)(13,15)(17,20,22,36)(18,25,26,43)(19,30,31,34)(21,35,23,37)(24,39,29,42)(27,33,28,44)(32,46,38,47)(40,48,41,45);(1,3,5,4)(2,6,7,8)(10,14)(13,15)(18,26)(19,31)(25,43)(27,28)(30,34)(32,38)(33,44)(46,47);(1,4,5,3)(2,6,7,8)(17,21,22,23)(18,27,26,28)(19,32,31,38)(20,35,36,37)(24,40,29,41)(25,33,43,44)(30,46,34,47)(39,48,42,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6896249);