#8824395

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	#48718	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3993 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)(3,11)(4,9)(5,10)(6,7)(8,12)(13,14)(15,16)(18,23)(21,29)(24,30)(28,32);(2,7)(5,15)(8,11)(9,13)(20,27)(24,30)(26,31)(28,32)(33,34,38,36)(35,41,43,39)(37,44,45,40)(42,46,48,47);(3,12)(4,14)(8,11)(9,13)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,3)(2,8)(4,10)(5,13)(6,12)(7,11)(9,15)(14,16)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(1,4)(2,9)(3,10)(5,11)(6,14)(7,13)(8,15)(12,16)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,36,38,34)(35,39,43,41)(37,44,45,40)(42,46,48,47);(1,5,6,15)(2,10,7,16)(3,13,12,9)(4,11,14,8)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=hi,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, 8731591);