#8843513

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	#53518	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 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)(17,18,23,26)(19,24,21,20)(22,30,31,25)(27,28,29,32)(33,34,39,42)(35,38,45,40)(36,46,41,37)(43,44,48,47);(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,18,23,26)(19,24,21,20)(22,30,31,25)(27,28,29,32)(33,35)(34,40)(36,43)(37,44)(38,42)(39,45)(41,48)(46,47);(2,3)(7,9)(8,10)(14,15)(17,19,23,21)(18,20,26,24)(22,27,31,29)(25,28,30,32)(33,36,39,41)(34,37,42,46)(35,43,45,48)(38,47,40,44);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,20)(18,19)(21,26)(22,28)(23,24)(25,27)(29,30)(31,32)(33,37,39,46)(34,41,42,36)(35,44,45,47)(38,43,40,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,21,23,19)(18,24,26,20)(22,29,31,27)(25,32,30,28)(33,34)(35,38)(36,37)(39,42)(40,45)(41,46)(43,47)(44,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,31)(18,25,26,30)(19,27,21,29)(20,28,24,32)(33,38)(34,35)(36,47)(37,43)(39,40)(41,44)(42,45)(46,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,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, 8750709);