#8720814

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

4015 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=ghi,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8628010);