#8504803

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	#58573	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=i,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8411999);