#8183731

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=g,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8090927);