#8745917

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8653113);