#8449357

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

4092 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(18,22)(23,28)(24,29)(30,32)(33,34)(35,45)(36,41)(37,42)(38,43)(39,44)(40,46)(47,48);(2,7)(3,10)(9,12)(11,16)(19,25)(23,28)(26,31)(30,32)(35,46)(36,47)(40,45)(41,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35,39,46)(34,40,44,45)(36,43,47,37)(38,41,42,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36)(34,41)(35,43)(37,46)(38,45)(39,47)(40,42)(44,48);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,37,39,43)(34,42,44,38)(35,36,46,47)(40,48,45,41);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,38,39,42)(34,43,44,37)(35,41,46,48)(36,40,47,45)

Polycyclic presentation:

2^9|c^2=gi,e^2=i,f^2=h,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, 8356553);