#8814022

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	#50800	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4169 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=ghi,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8721218);