#10181240

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	#51126	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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)(3,13,14,8)(4,5,11,10)(6,16,15,12)(34,40)(36,46)(41,43)(44,45);(1,3)(2,8)(4,15)(5,16)(6,11)(7,14)(9,13)(10,12)(17,18,23,21)(19,26,20,24)(22,25,30,29)(27,32,28,31)(33,34)(35,41)(36,42)(37,43)(38,44)(39,40)(45,48)(46,47);(2,9)(3,14)(4,11)(12,16)(17,19,23,20)(18,24,21,26)(22,27,30,28)(25,31,29,32)(33,35,39,37)(34,41,40,43)(36,45,46,44)(38,42,48,47);(1,4,7,11)(2,10,9,5)(3,15,14,6)(8,12,13,16)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32)(33,36)(34,42)(35,45)(37,44)(38,43)(39,46)(40,47)(41,48);(1,5,7,10)(2,11,9,4)(3,12,14,16)(6,8,15,13)(17,21,23,18)(19,26,20,24)(22,25,30,29)(27,31,28,32)(33,37,39,35)(34,43,40,41)(36,45,46,44)(38,42,48,47);(1,6,7,15)(2,12,9,16)(3,11,14,4)(5,8,10,13)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38,39,48)(34,44,40,45)(35,42,37,47)(36,43,46,41)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gi,d^2=i,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, 10088436);