#8843593

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	#53666	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3968 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=hi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8750789);