#8901572

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	#40346	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4050 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)(33,34,39,40)(35,42,36,41)(37,43,47,45)(38,46,48,44);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,32)(28,30)(34,40)(37,47)(41,42)(44,46);(5,13)(8,10)(11,16)(14,15)(18,22)(23,29)(24,30)(31,32)(33,35,39,36)(34,41,40,42)(37,44,47,46)(38,43,48,45);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,19,21,26)(18,23,22,29)(20,27,28,25)(24,32,30,31)(33,36,39,35)(34,42,40,41)(37,44,47,46)(38,43,48,45);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20,21,28)(18,24,22,30)(19,25,26,27)(23,31,29,32)(33,37,39,47)(34,43,40,45)(35,44,36,46)(38,41,48,42);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,32)(28,30)(33,38)(34,44)(35,45)(36,43)(37,41)(39,48)(40,46)(42,47)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8808768);