#8183746

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	#52636	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4154 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)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,18,22,24)(19,23,26,20)(21,29,30,25)(27,28,32,31)(33,34)(35,42)(36,45)(37,46)(38,40)(39,41)(43,44)(47,48);(2,3)(7,9)(8,10)(14,15)(17,19)(18,23)(20,24)(21,27)(22,26)(25,31)(28,29)(30,32)(33,35,39,38)(34,40,41,42)(36,43,46,48)(37,44,45,47);(5,13)(8,10)(11,16)(14,15)(18,24)(19,26)(25,29)(27,32)(33,36)(34,37)(35,43)(38,48)(39,46)(40,44)(41,45)(42,47);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,22,23)(18,19,24,26)(21,28,30,31)(25,27,29,32)(33,37)(34,36)(35,44)(38,47)(39,45)(40,43)(41,46)(42,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21,22,30)(18,25,24,29)(19,27,26,32)(20,28,23,31)(33,37)(34,36)(35,44)(38,47)(39,45)(40,43)(41,46)(42,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,19)(18,23)(20,24)(21,27)(22,26)(25,31)(28,29)(30,32)(33,38,39,35)(34,40,41,42)(36,43,46,48)(37,47,45,44)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8090942);