#8912187

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

4220 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=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, 8819383);