#8462119

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

4195 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8369315);