#10181248

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	#46600	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) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 10088444);