#8621847

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	#40842	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4129 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8529043);