#8159860

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	#52391	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4006 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=hi,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^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, 8067056);