#9937708

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	#55306	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 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)(3,5)(4,13)(6,15)(7,8)(9,14)(10,12)(11,16)(17,18)(19,22)(20,27)(21,24)(23,29)(25,32)(26,31)(28,30)(33,34,38,40)(35,36,41,43)(37,39,46,48)(42,44,45,47);(1,3)(2,5)(4,6)(7,11)(8,16)(9,10)(12,14)(13,15)(17,18,21,24)(19,22,26,31)(20,23,29,27)(25,30,28,32)(35,41)(36,43)(42,45)(44,47);(2,8)(3,11)(4,14)(10,15)(17,18,21,24)(19,22,26,31)(20,23,29,27)(25,30,28,32)(33,35)(34,36)(37,42)(38,41)(39,44)(40,43)(45,46)(47,48);(1,4)(2,9)(3,12)(5,15)(6,11)(7,14)(8,13)(10,16)(17,18)(19,22)(20,23)(21,24)(25,30)(26,31)(27,29)(28,32)(33,36)(34,35)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47);(1,5)(2,3)(4,15)(6,13)(7,16)(8,11)(9,12)(10,14)(17,19,21,26)(18,22,24,31)(20,28,29,25)(23,32,27,30)(33,35)(34,36)(37,45)(38,41)(39,47)(40,43)(42,46)(44,48);(1,6,7,12)(2,10,8,15)(3,4,11,14)(5,9,16,13)(17,20)(18,23)(19,25)(21,29)(22,30)(24,27)(26,28)(31,32)(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=g,c^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, 9844904);