#8343531

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

3921 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8250727);