#8755110

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3845 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662306);