#8093463

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	#52653	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3859 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000659);