#8806742

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

4054 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)(3,8,11,5)(4,9,14,15)(10,16,12,13)(17,18)(19,25)(20,27)(21,29)(22,24)(23,30)(26,28)(31,32)(33,34,40,46)(35,43,37,41)(36,45,39,42)(38,44,48,47);(2,7)(5,8)(9,15)(13,16)(17,18,22,24)(19,25,27,20)(21,23,30,29)(26,31,32,28)(33,35,40,37)(34,41,46,43)(36,47,39,44)(38,45,48,42);(1,3,6,11)(2,8,7,5)(4,12,14,10)(9,13,15,16)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)(29,31)(30,32)(33,36,40,39)(34,42,46,45)(35,47,37,44)(38,43,48,41);(1,3)(2,5)(4,10)(6,11)(7,8)(9,13)(12,14)(15,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,37,40,35)(34,43,46,41)(36,47,39,44)(38,45,48,42);(1,4)(2,9)(3,10)(5,13)(6,14)(7,15)(8,16)(11,12)(17,18)(19,20)(21,23)(22,24)(25,27)(26,28)(29,30)(31,32)(33,38,40,48)(34,44,46,47)(35,42,37,45)(36,41,39,43);(1,5)(2,3)(4,13)(6,8)(7,11)(9,10)(12,15)(14,16)(17,21,22,30)(18,23,24,29)(19,26,27,32)(20,28,25,31)(33,39,40,36)(34,45,46,42)(35,47,37,44)(38,43,48,41)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=hi,e^2=hi,f^2=h,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, 8713938);