#8127469

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

3926 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)(3,12)(4,5)(6,9)(7,8)(10,11)(13,16)(14,15)(18,21)(19,27)(20,30)(26,32)(33,34,40,45)(35,44,39,41)(36,38,43,47)(37,46,48,42);(2,8)(4,11)(9,12)(13,15)(17,18,23,21)(19,28,27,24)(20,31,30,25)(22,26,29,32)(33,35,40,39)(34,41,45,44)(36,46,43,42)(37,47,48,38);(1,3,7,6)(2,9,8,12)(4,15,11,13)(5,16,10,14)(17,19)(18,24)(20,29)(21,28)(22,30)(23,27)(25,32)(26,31)(33,36)(34,38)(35,46)(37,41)(39,42)(40,43)(44,48)(45,47);(1,4,7,11)(2,10,8,5)(3,13,6,15)(9,16,12,14)(17,20,23,30)(18,25,21,31)(19,29,27,22)(24,32,28,26)(33,37,40,48)(34,42,45,46)(35,47,39,38)(36,44,43,41);(1,5,7,10)(2,11,8,4)(3,14,6,16)(9,15,12,13)(17,21,23,18)(19,28,27,24)(20,31,30,25)(22,32,29,26)(33,38)(34,43)(35,37)(36,45)(39,48)(40,47)(41,42)(44,46);(1,6,7,3)(2,12,8,9)(4,15,11,13)(5,16,10,14)(17,22,23,29)(18,26,21,32)(19,30,27,20)(24,31,28,25)(33,39,40,35)(34,44,45,41)(36,42,43,46)(37,38,48,47)

Polycyclic presentation:

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

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, 8034665);