#9717467

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3789 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9624663);