#8840313

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	#50849	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4131 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:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=i,f^2=h,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, 8747509);