#8399501

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	#45663	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4061 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)(3,10)(4,8)(5,14)(6,12)(7,9)(11,15)(13,16)(17,18,23,26)(19,22,25,32)(20,21,24,31)(27,28,29,30)(33,34)(35,40)(36,37)(38,41)(39,42)(43,47)(44,45)(46,48);(1,3,7,4)(2,8,9,10)(5,13,15,12)(6,11,16,14)(17,19)(18,22)(20,29)(21,30)(23,25)(24,27)(26,32)(28,31)(33,34,39,42)(35,40,36,37)(38,41,46,48)(43,47,44,45);(2,9)(6,16)(8,10)(11,14)(19,25)(22,32)(27,29)(28,30)(33,35,39,36)(34,40,42,37)(38,44,46,43)(41,45,48,47);(1,4,7,3)(2,10,9,8)(5,12,15,13)(6,14,16,11)(17,20)(18,21)(19,27)(22,28)(23,24)(25,29)(26,31)(30,32)(33,36,39,35)(34,40,42,37)(38,44,46,43)(41,47,48,45);(1,5,7,15)(2,11,9,14)(3,12,4,13)(6,10,16,8)(17,21)(18,24)(19,28)(20,26)(22,29)(23,31)(25,30)(27,32)(33,37,39,40)(34,36,42,35)(38,45,46,47)(41,44,48,43);(1,6,7,16)(2,12,9,13)(3,11,4,14)(5,10,15,8)(17,22,23,32)(18,25,26,19)(20,28,24,30)(21,29,31,27)(33,38,39,46)(34,41,42,48)(35,43,36,44)(37,45,40,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=g,e^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, 8306697);