#6993350

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	#38035	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1556 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)(3,7)(4,6)(5,8)(17,18,22,27,23,28,19,24)(20,31,29,40,33,42,37,25)(21,34,35,38,36,26,30,41)(32,45,43,39,47,44,46,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19,23,22)(18,24,28,27)(20,29,33,37)(21,30,36,35)(25,31,40,42)(26,38,34,41)(32,43,47,46)(39,44,48,45);(3,8)(5,7)(9,11,14,10)(12,15,13,16)(17,20)(18,25)(19,29)(21,32)(22,37)(23,33)(24,31)(26,39)(27,42)(28,40)(30,43)(34,48)(35,46)(36,47)(38,44)(41,45);(2,6)(5,7)(9,12,14,13)(10,15,11,16)(17,21,23,36)(18,26,28,34)(19,30,22,35)(20,32,33,47)(24,38,27,41)(25,39,40,48)(29,43,37,46)(31,44,42,45);(9,13,14,12)(10,15,11,16)(17,19,23,22)(18,24,28,27)(20,29,33,37)(21,35,36,30)(25,31,40,42)(26,41,34,38)(32,46,47,43)(39,45,48,44)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fh,d^2=fhi,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, 6900546);