#9453090

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

4050 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9360286);