#8630888

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	#35053	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4095 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,d^2=g,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8538084);