#8746534

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	#40126	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4145 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)(19,23)(20,25)(21,26)(22,27)(24,28)(29,31)(30,32)(33,34,39,38)(35,44,46,40)(36,42,47,41)(37,48,45,43);(1,2,5,6)(3,7,11,9)(4,8,14,12)(10,15,13,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,27)(25,31)(26,32)(33,35)(34,40)(36,37)(38,44)(39,46)(41,48)(42,43)(45,47);(2,6)(7,9)(8,12)(15,16)(19,28)(23,24)(29,32)(30,31)(33,36,39,47)(34,41,38,42)(35,37,46,45)(40,48,44,43);(1,2)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,19)(18,24)(20,29)(21,31)(22,28)(23,27)(25,30)(26,32)(33,36)(34,42)(35,45)(37,46)(38,41)(39,47)(40,48)(43,44);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,20)(18,25)(19,29)(21,27)(22,26)(23,31)(24,30)(28,32)(33,37,39,45)(34,43,38,48)(35,36,46,47)(40,42,44,41);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,21,22,25)(18,26,27,20)(19,30,28,31)(23,32,24,29)(33,38,39,34)(35,44,46,40)(36,42,47,41)(37,43,45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8653730);