#9794297

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=h,e^2=gh,f^2=gh,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, 9701493);