#9937758

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3888 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)(3,7)(4,10)(5,13)(6,8)(9,12)(11,16)(14,15)(17,18)(19,28)(20,21)(22,32)(23,27)(24,30)(25,31)(26,29)(33,34,38,40)(35,36,41,43)(37,39,46,48)(42,44,45,47);(1,3,6,12)(2,7,8,9)(4,13,15,11)(5,14,16,10)(17,19)(18,24)(20,22)(21,26)(23,30)(25,32)(27,28)(29,31)(35,41)(36,43)(42,45)(44,47);(2,8)(3,12)(10,14)(11,13)(17,20,23,31)(18,25,27,21)(19,22,30,29)(24,32,28,26)(33,35)(34,36)(37,42)(38,41)(39,44)(40,43)(45,46)(47,48);(1,3)(2,9)(4,13)(5,14)(6,12)(7,8)(10,16)(11,15)(17,20)(18,21)(19,29)(22,30)(23,31)(24,32)(25,27)(26,28)(33,36)(34,35)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47);(1,4)(2,10)(3,13)(5,7)(6,15)(8,14)(9,16)(11,12)(17,21)(18,20)(19,26)(22,24)(23,25)(27,31)(28,29)(30,32)(33,35)(34,36)(37,45)(38,41)(39,47)(40,43)(42,46)(44,48);(1,5,6,16)(2,11,8,13)(3,14,12,10)(4,9,15,7)(17,22,23,29)(18,26,27,32)(19,20,30,31)(21,28,25,24)(33,37)(34,39)(35,42)(36,44)(38,46)(40,48)(41,45)(43,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9844954);