#8533520

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

4250 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)(3,11)(4,13)(5,10)(6,7)(8,12)(9,14)(15,16)(17,18)(19,20)(21,25)(22,23)(24,26)(27,28)(29,32)(30,31)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(2,7)(3,12)(9,13)(10,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,31)(30,32)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(2,7)(4,14)(8,11)(10,16)(18,23)(19,26)(25,31)(27,32)(33,35,39,36)(34,41,37,40)(38,43,46,44)(42,48,45,47);(1,3)(2,8)(4,10)(5,9)(6,12)(7,11)(13,15)(14,16)(17,20,22,24)(18,19,23,26)(21,29,30,28)(25,32,31,27)(33,36,39,35)(34,41,37,40)(38,43,46,44)(42,47,45,48);(1,4,6,14)(2,9,7,13)(3,10,12,16)(5,11,15,8)(17,19)(18,24)(20,23)(21,27)(22,26)(25,29)(28,31)(30,32)(33,37,39,34)(35,41,36,40)(38,45,46,42)(43,48,44,47);(1,5)(2,10)(3,13)(4,11)(6,15)(7,16)(8,14)(9,12)(17,21,22,30)(18,25,23,31)(19,27,26,32)(20,28,24,29)(33,38,39,46)(34,42,37,45)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440716);