#10196265

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	#48174	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3980 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gi,d^2=h,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, 10103461);