#6970472

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907909
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1436 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)(19,29,32,24)(20,33,35,40)(21,28,23,26)(25,34,41,36)(30,44,46,47)(31,39,37,42)(38,45,43,48);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19,22,32)(18,24,27,29)(20,30,35,46)(21,37,23,31)(25,38,41,43)(26,42,28,39)(33,47,40,44)(34,48,36,45);(3,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20,21,34,22,35,23,36)(18,25,26,40,27,41,28,33)(19,30,37,48,32,46,31,45)(24,38,42,44,29,43,39,47);(2,6)(5,7)(10,13)(14,16)(17,21,22,23)(18,26,27,28)(19,31,32,37)(20,34,35,36)(24,39,29,42)(25,40,41,33)(30,45,46,48)(38,47,43,44);(9,11,12,15)(10,14,13,16)(19,32)(24,29)(30,46)(31,37)(38,43)(39,42)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfhi

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, 6877668);