#10196109

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	#41260	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4066 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,e^2=g,f^2=gi,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, 10103305);