#8324310

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	#57981	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4168 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)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,18,23,26)(19,22,25,31)(20,21,24,30)(27,28,29,32)(33,34,40,44)(35,45,46,39)(36,42,37,41)(38,48,47,43);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,19)(18,22)(20,27)(21,28)(23,25)(24,29)(26,31)(30,32)(33,35,40,46)(34,39,44,45)(36,38,37,47)(41,48,42,43);(2,6)(7,9)(8,12)(15,16)(19,25)(22,31)(27,29)(28,32)(33,36,40,37)(34,41,44,42)(35,38,46,47)(39,48,45,43);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,20)(18,21)(19,27)(22,28)(23,24)(25,29)(26,30)(31,32)(33,37,40,36)(34,42,44,41)(35,38,46,47)(39,48,45,43);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,16)(11,13)(17,21)(18,24)(19,28)(20,26)(22,29)(23,30)(25,32)(27,31)(33,38)(34,43)(35,37)(36,46)(39,41)(40,47)(42,45)(44,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,14)(13,16)(17,22,23,31)(18,25,26,19)(20,28,24,32)(21,29,30,27)(33,39,40,45)(34,35,44,46)(36,43,37,48)(38,41,47,42)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8231506);