#6989152

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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	#38032	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1523 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)(17,18,23,24,22,28,19,27)(20,31,29,25,34,38,32,40)(21,26,35,37,36,41,30,42)(33,44,43,39,46,47,45,48);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,23)(18,24,28,27)(20,32,34,29)(21,30,36,35)(25,38,40,31)(26,37,41,42)(33,45,46,43)(39,47,48,44);(1,2)(3,8)(4,6)(5,7)(11,16)(13,15)(17,20,22,34)(18,25,28,40)(19,29,23,32)(21,33,36,46)(24,31,27,38)(26,39,41,48)(30,43,35,45)(37,44,42,47);(1,3,5,4)(2,6,7,8)(10,14)(13,15)(17,21)(18,26)(19,30)(20,33)(22,36)(23,35)(24,37)(25,39)(27,42)(28,41)(29,43)(31,44)(32,45)(34,46)(38,47)(40,48);(1,4,5,3)(2,6,7,8)(17,19,22,23)(18,27,28,24)(20,29,34,32)(21,35,36,30)(25,38,40,31)(26,37,41,42)(33,45,46,43)(39,44,48,47)

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=dfi,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

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, 6896348);