#8821770

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

4052 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=gh,e^2=ghi,f^2=hi,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, 8728966);