#6993318

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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

1563 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)(20,31,29,39,33,42,36,25)(21,26,34,40,35,41,30,37)(32,44,43,48,46,47,45,38);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19,23,22)(18,24,28,27)(20,29,33,36)(21,30,35,34)(25,31,39,42)(26,37,41,40)(32,43,46,45)(38,44,48,47);(3,8)(5,7)(9,11,14,10)(12,15,13,16)(17,20)(18,25)(19,29)(21,32)(22,36)(23,33)(24,31)(26,38)(27,42)(28,39)(30,43)(34,45)(35,46)(37,44)(40,47)(41,48);(2,6)(5,7)(9,12,14,13)(10,15,11,16)(17,21,23,35)(18,26,28,41)(19,30,22,34)(20,32,33,46)(24,37,27,40)(25,38,39,48)(29,43,36,45)(31,44,42,47);(9,13,14,12)(10,15,11,16)(17,19,23,22)(18,24,28,27)(20,29,33,36)(21,34,35,30)(25,31,39,42)(26,40,41,37)(32,45,46,43)(38,47,48,44)

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6900514);