#8757453

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	#45986	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3935 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8664649);