#8295547

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	#52530	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4149 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=gi,e^2=gh,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, 8202743);