#8919501

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	#57142	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4249 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8826697);