#8139485

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

4066 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)(3,8)(4,6)(5,12)(7,11)(9,13)(10,14)(15,16)(17,18,21,25)(19,23,26,22)(20,28,29,24)(27,30,32,31)(33,34,39,44)(35,45,46,40)(36,42,37,41)(38,47,48,43);(1,3)(2,7)(4,8)(5,10)(6,11)(9,14)(12,15)(13,16)(19,26)(22,23)(27,32)(30,31)(33,35,39,46)(34,40,44,45)(36,43,37,47)(38,42,48,41);(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32)(33,36,39,37)(34,41,44,42)(35,43,46,47)(38,45,48,40);(1,4,6,2)(3,8,11,7)(5,12,13,9)(10,15,16,14)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,37,39,36)(34,42,44,41)(35,43,46,47)(38,45,48,40);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32)(33,38)(34,43)(35,42)(36,45)(37,40)(39,48)(41,46)(44,47);(1,5,6,13)(2,9,4,12)(3,10,11,16)(7,14,8,15)(17,20,21,29)(18,24,25,28)(19,27,26,32)(22,30,23,31)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=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, 8046681);