#8824747

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	#55007	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4147 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=h,e^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, 8731943);