#8472385

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

4097 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8379581);