#9794252

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3737 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9701448);