#8757136

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

4138 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)(3,9)(4,12)(5,10)(6,7)(8,14)(11,13)(15,16)(17,18,23,25)(19,24,29,32)(20,26,28,22)(21,31,30,27)(35,41)(38,42)(43,47)(46,48);(2,7)(5,9)(8,12)(13,15)(17,19)(18,24)(20,30)(21,28)(22,31)(23,29)(25,32)(26,27)(33,34)(35,42)(36,44)(37,40)(38,41)(39,45)(43,46)(47,48);(2,7)(3,10)(8,12)(11,16)(18,25)(19,29)(20,28)(27,31)(33,34,37,40)(35,38,41,42)(36,39,45,44)(43,46,47,48);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,15)(14,16)(17,20,23,28)(18,26,25,22)(19,30,29,21)(24,27,32,31)(33,34)(35,38)(36,39)(37,40)(41,42)(43,46)(44,45)(47,48);(1,4,6,14)(2,8,7,12)(3,11,10,16)(5,13,9,15)(17,21,23,30)(18,27,25,31)(19,28,29,20)(22,24,26,32)(33,35,37,41)(34,38,40,42)(36,43,45,47)(39,46,44,48);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,16)(14,15)(17,22,23,26)(18,28,25,20)(19,27,29,31)(21,24,30,32)(33,36)(34,39)(35,43)(37,45)(38,46)(40,44)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=gh,c^2=ghi,d^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664332);