#8824445

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

4236 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:

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

Polycyclic presentation:

2^9|a^2=i,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, 8731641);