#10196117

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4161 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)(3,6,4,8)(5,10,14,16)(11,13,12,15)(18,23)(19,26)(25,32)(27,28)(33,34,38,41)(35,37,43,39)(36,40,44,47)(42,45,48,46);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,31)(30,32)(36,44)(40,47)(42,48)(45,46);(2,9)(3,4)(10,16)(11,12)(33,35,38,43)(34,39,41,37)(36,42,44,48)(40,46,47,45);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,19,22,26)(18,24,23,21)(20,28,30,27)(25,31,32,29)(33,35)(34,37)(36,42)(38,43)(39,41)(40,45)(44,48)(46,47);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,13,8,15)(17,20,22,30)(18,25,23,32)(19,27,26,28)(21,31,24,29)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,27,32,28)(33,37)(34,35)(36,45)(38,39)(40,42)(41,43)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=g,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, 10103313);