#8824117

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3923 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)(4,11)(5,7)(8,13)(9,15)(10,16)(12,14)(17,18,23,20)(19,26,25,22)(21,24,29,28)(27,32,31,30)(33,34,39,40)(35,43,46,37)(36,45,48,41)(38,44,47,42);(1,3,5,9)(2,6,7,15)(4,12,13,10)(8,16,11,14)(21,29)(24,28)(27,31)(30,32)(34,40)(37,43)(38,47)(41,45);(2,7)(3,9)(8,11)(10,12)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,32)(29,31)(33,35,39,46)(34,37,40,43)(36,42,48,44)(38,45,47,41);(1,2)(3,6)(4,8)(5,7)(9,15)(10,14)(11,13)(12,16)(17,20,23,18)(19,26,25,22)(21,28,29,24)(27,32,31,30)(33,36,39,48)(34,41,40,45)(35,44,46,42)(37,47,43,38);(1,3)(2,6)(4,10)(5,9)(7,15)(8,14)(11,16)(12,13)(17,21,23,29)(18,24,20,28)(19,27,25,31)(22,30,26,32)(33,37)(34,35)(36,47)(38,48)(39,43)(40,46)(41,44)(42,45);(1,4)(2,8)(3,10)(5,13)(6,14)(7,11)(9,12)(15,16)(17,22,23,26)(18,25,20,19)(21,30,29,32)(24,31,28,27)(33,38,39,47)(34,42,40,44)(35,45,46,41)(36,37,48,43)

Polycyclic presentation:

2^9|a^2=hi,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, 8731313);