#10181217

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	#54647	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3910 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=gi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10088413);