#8824440

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	#48071	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8731636);