#8720508

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8627704);