#8464372

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

4153 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8371568);