#10183141

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	#37221	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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)(3,8,11,15)(4,6,13,16)(5,10,12,14)(17,18,23,25)(19,28,22,24)(20,29,30,26)(21,27,32,31)(34,38)(36,44)(39,41)(42,48);(1,3,7,11)(2,8,9,15)(4,12,13,5)(6,14,16,10)(17,19,23,22)(18,24,25,28)(20,27,30,31)(21,26,32,29)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(2,9)(3,11)(4,13)(10,14)(18,25)(20,30)(24,28)(27,31)(33,34,37,38)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,4)(2,6)(3,12)(5,11)(7,13)(8,14)(9,16)(10,15)(17,20,23,30)(18,26,25,29)(19,27,22,31)(21,24,32,28)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,5,7,12)(2,10,9,14)(3,13,11,4)(6,8,16,15)(17,21,23,32)(18,27,25,31)(19,29,22,26)(20,28,30,24)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,6)(2,4)(3,14)(5,15)(7,16)(8,12)(9,13)(10,11)(17,22,23,19)(18,28,25,24)(20,31,30,27)(21,29,32,26)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=ghi,d^2=hi,e^2=gi,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, 10090337);