#8813951

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3851 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=h,e^2=gh,f^2=hi,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, 8721147);