#9685524

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

4100 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,23,26)(19,21,25,31)(20,22,30,24)(27,32,28,29)(33,34,40,44)(35,43,39,41)(36,37,38,42)(45,48,47,46);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,23,25)(18,21,26,31)(20,28,30,27)(22,29,24,32)(33,35,40,39)(34,41,44,43)(36,45,38,47)(37,46,42,48);(2,3)(7,9)(8,10)(14,15)(17,20,23,30)(18,24,26,22)(19,27,25,28)(21,29,31,32)(33,36,40,38)(34,42,44,37)(35,45,39,47)(41,48,43,46);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,22)(19,28)(21,29)(23,30)(24,26)(25,27)(31,32)(33,37,40,42)(34,36,44,38)(35,46,39,48)(41,45,43,47);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21,23,31)(18,25,26,19)(20,29,30,32)(22,27,24,28)(33,38,40,36)(34,42,44,37)(35,45,39,47)(41,46,43,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22)(18,20)(19,29)(21,28)(23,24)(25,32)(26,30)(27,31)(33,39,40,35)(34,43,44,41)(36,47,38,45)(37,48,42,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9592720);