#8840200

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	#53949	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=hi,e^2=gh,f^2=g,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, 8747396);