#8824113

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	#48602	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3989 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=hi,e^2=ghi,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, 8731309);