#8323523

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

4126 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=h,e^2=g,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, 8230719);