#8824451

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4126 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)(13,15)(14,16)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(34,38)(36,43)(39,44)(42,47);(2,7)(5,14)(8,11)(9,15)(17,18,22,19)(20,25,27,23)(21,26,29,24)(28,30,32,31)(35,41)(39,44)(42,47)(46,48);(3,12)(4,13)(8,11)(9,15)(18,19)(20,27)(24,26)(28,32)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,3,6,12)(2,8,7,11)(4,10,13,16)(5,9,14,15)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30);(1,4,6,13)(2,9,7,15)(3,10,12,16)(5,8,14,11)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,35,37,41)(34,39,38,44)(36,42,43,47)(40,46,45,48);(1,5,6,14)(2,10,7,16)(3,9,12,15)(4,8,13,11)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=gi,e^2=ghi,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, 8731647);