#8773036

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	#41530	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3918 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)(3,11)(4,14)(5,15)(6,10)(7,13)(8,12)(9,16)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,34,38,39)(35,41,44,40)(36,37,45,46)(42,47,48,43);(1,3)(2,7)(4,5)(6,13)(8,15)(9,14)(10,11)(12,16)(17,18,23,20)(19,24,22,26)(21,28,30,25)(27,32,29,31)(34,39)(36,45)(40,41)(42,48);(1,2,6,10)(3,7,13,11)(4,14,12,8)(5,9,16,15)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,34)(35,40)(36,37)(38,39)(41,44)(42,43)(45,46)(47,48);(1,4,6,12)(2,8,10,14)(3,5,13,16)(7,15,11,9)(17,20,23,18)(19,26,22,24)(21,28,30,25)(27,32,29,31)(33,35,38,44)(34,40,39,41)(36,42,45,48)(37,43,46,47);(1,2)(3,11)(4,8)(5,9)(6,10)(7,13)(12,14)(15,16)(17,21,23,30)(18,25,20,28)(19,27,22,29)(24,31,26,32)(33,36)(34,37)(35,42)(38,45)(39,46)(40,43)(41,47)(44,48);(1,5)(2,9)(3,12)(4,13)(6,16)(7,8)(10,15)(11,14)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,37)(34,36)(35,43)(38,46)(39,45)(40,42)(41,48)(44,47)

Polycyclic presentation:

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

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, 8680232);