#6711953

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#42806	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1515 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,25)(19,27,30,23)(20,31,33,24)(22,26,35,42)(28,38,46,45)(29,43,36,39)(32,40,47,48)(34,44,37,41);(10,13)(11,15)(17,19,21,30)(18,23,25,27)(20,32,33,47)(22,36,35,29)(24,40,31,48)(26,43,42,39)(28,37,46,34)(38,44,45,41);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,28)(21,33)(22,37)(23,38)(25,31)(26,44)(27,45)(29,47)(30,46)(32,36)(34,35)(39,48)(40,43)(41,42);(10,13)(14,16)(20,33)(24,31)(28,46)(32,47)(34,37)(38,45)(40,48)(41,44);(1,3,4,7)(2,6,5,8)(19,29)(22,35)(23,39)(26,42)(27,43)(28,47)(30,36)(32,46)(34,37)(38,48)(40,45)(41,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6619149);