#8824387

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	#38761	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4016 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,d^2=gi,e^2=i,f^2=gi,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, 8731583);