#8093459

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	#52360	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3855 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)(19,27)(20,21)(22,24)(23,26)(25,32)(28,29)(30,31)(33,34)(35,43)(36,38)(37,42)(39,40)(41,47)(44,45)(46,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,23,22)(18,24,26,27)(20,30,25,28)(21,29,32,31)(34,40)(36,42)(41,43)(44,48);(2,6)(7,9)(8,13)(14,15)(17,20,23,25)(18,21,26,32)(19,28,22,30)(24,31,27,29)(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)(18,25)(19,29)(20,26)(22,31)(23,32)(24,28)(27,30)(33,36)(34,37)(35,44)(38,40)(39,42)(41,45)(43,46)(47,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(33,37,39,38)(34,42,40,36)(35,45,47,46)(41,48,43,44);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,19)(18,24,26,27)(20,30,25,28)(21,31,32,29)(33,38,39,37)(34,36,40,42)(35,46,47,45)(41,44,43,48)

Polycyclic presentation:

2^9|b^2=gh,c^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, 8000655);