#8186906

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	#34967	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(33,34,38,42)(35,40,36,39)(37,41,45,48)(43,47,44,46);(2,3)(7,9)(8,10)(14,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,35,38,36)(34,39,42,40)(37,44,45,43)(41,47,48,46);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21,22,30)(18,24,25,29)(19,27,20,28)(23,31,26,32)(33,36,38,35)(34,40,42,39)(37,44,45,43)(41,47,48,46);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,37,38,45)(34,41,42,48)(35,43,36,44)(39,46,40,47)

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=i,e^2=h,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, 8094102);