#8993915

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

4194 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=ghi,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,e^c=egi,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, 8901111);