#9937704

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3895 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)(5,8)(9,11)(10,14)(12,15)(13,16)(17,18,23,24)(19,26,21,20)(22,30,31,25)(27,28,29,32)(33,34)(35,37)(36,45)(38,47)(39,40)(41,46)(42,44)(43,48);(1,2,5,8)(3,9,11,6)(4,7,13,16)(10,15,12,14)(17,19,23,21)(18,20,24,26)(22,27,31,29)(25,28,30,32)(33,35)(34,37)(36,38)(39,43)(40,48)(41,42)(44,46)(45,47);(18,24)(20,26)(25,30)(28,32)(34,40)(35,43)(36,46)(42,47);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,19)(21,24)(22,28)(23,26)(25,27)(29,30)(31,32)(33,36)(34,41)(35,44)(37,47)(38,43)(39,46)(40,45)(42,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,21,23,19)(18,20,24,26)(22,27,31,29)(25,32,30,28)(33,37)(34,35)(36,47)(38,45)(39,48)(40,43)(41,44)(42,46);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,30)(26,32)(33,38,39,44)(34,42,40,47)(35,36,43,46)(37,41,48,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,e^2=g,f^2=gh,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, 9844900);