#8227358

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

3862 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)(19,26)(20,23)(21,28)(22,25)(24,30)(27,29)(31,32)(33,34,39,43)(35,36,46,40)(37,41,38,42)(44,47,45,48);(1,2,6,7)(3,5,9,14)(4,11,13,8)(10,16,15,12)(18,20)(19,25)(24,28)(27,31)(33,35,39,46)(34,40,43,36)(37,44,38,45)(41,48,42,47);(2,7)(5,14)(8,11)(12,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,35)(34,36)(37,44)(38,45)(39,46)(40,43)(41,47)(42,48);(1,3,6,9)(2,5,7,14)(4,10,13,15)(8,12,11,16)(17,20,23,18)(19,26,25,22)(21,28,30,24)(27,32,31,29)(33,36)(34,35)(37,47)(38,48)(39,40)(41,44)(42,45)(43,46);(1,4,6,13)(2,8,7,11)(3,10,9,15)(5,12,14,16)(17,21,23,30)(18,24,20,28)(19,27,25,31)(22,29,26,32)(33,37,39,38)(34,41,43,42)(35,44,46,45)(36,47,40,48);(1,5)(2,9)(3,7)(4,12)(6,14)(8,15)(10,11)(13,16)(17,22,23,26)(18,25,20,19)(21,29,30,32)(24,31,28,27)(33,38,39,37)(34,42,43,41)(35,45,46,44)(36,48,40,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,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, 8134554);