#8824432

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

4170 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=hi,e^2=i,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, 8731628);