#10506330

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	#41875	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3926 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10413526);