#8504779

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

3945 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)(3,9)(4,14)(5,13)(6,11)(7,8)(10,15)(12,16)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(2,8)(5,15)(9,14)(11,12)(20,28)(25,30)(27,29)(31,32)(34,38)(39,41)(40,45)(46,47);(1,3,7,4)(2,9,8,14)(5,12,15,11)(6,13,16,10)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,4,7,3)(2,9,8,14)(5,11,15,12)(6,13,16,10)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,31)(26,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5)(2,10)(3,12)(4,11)(6,9)(7,15)(8,13)(14,16)(17,21,23,19)(18,26,22,24)(20,29,28,27)(25,32,30,31);(1,6)(2,11)(3,13)(4,10)(5,9)(7,16)(8,12)(14,15)(17,22,23,18)(19,26,21,24)(20,30,28,25)(27,32,29,31)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=gi,e^2=h,f^2=hi,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, 8411975);