#8944435

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	#48271	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4100 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

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, 8851631);