#8260691

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4019 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)(3,9,11,7)(4,8,12,13)(10,15,16,14)(17,18,23,26)(19,24,30,25)(20,31,21,22)(27,29,28,32)(33,34,40,44)(35,45,47,41)(36,37,38,42)(39,43,48,46);(2,6)(7,9)(8,13)(14,15)(17,19,23,30)(18,24,26,25)(20,28,21,27)(22,29,31,32)(33,35,40,47)(34,41,44,45)(36,46,38,43)(37,48,42,39);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,23,21)(18,22,26,31)(19,27,30,28)(24,32,25,29)(33,36,40,38)(34,42,44,37)(35,46,47,43)(39,45,48,41);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,21,23,20)(18,22,26,31)(19,28,30,27)(24,32,25,29)(33,37)(34,36)(35,39)(38,44)(40,42)(41,43)(45,46)(47,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19)(18,25)(20,27)(21,28)(22,29)(23,30)(24,26)(31,32)(33,38,40,36)(34,37,44,42)(35,46,47,43)(39,45,48,41);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,22)(18,20)(19,29)(21,26)(23,31)(24,28)(25,27)(30,32)(33,39)(34,43)(35,37)(36,41)(38,45)(40,48)(42,47)(44,46)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gi,d^2=gh,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=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, 8167887);