#8227363

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3837 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8134559);