#10042533

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3882 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=i,e^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=eh,d^c=dh,e^c=egi,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, 9949729);