#8399485

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	#34962	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4079 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8306681);