#10181235

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3936 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gi,d^2=i,e^2=hi,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, 10088431);