#8339889

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3903 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)(3,11)(4,14)(5,10)(6,7)(8,12)(9,15)(13,16)(17,18,23,22)(19,24,26,20)(21,29,31,25)(27,28,32,30)(33,34,39,36)(35,40,38,42)(37,41,46,44)(43,47,45,48);(2,7)(3,12)(9,14)(10,13)(17,19)(18,24)(20,22)(21,27)(23,26)(25,30)(28,29)(31,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(2,7)(5,16)(8,11)(9,14)(18,22)(19,26)(25,29)(27,32)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20)(18,19)(21,30)(22,26)(23,24)(25,32)(27,29)(28,31)(33,36,39,34)(35,42,38,40)(37,41,46,44)(43,47,45,48);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,21,23,31)(18,25,22,29)(19,27,26,32)(20,28,24,30)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,22,23,18)(19,20,26,24)(21,29,31,25)(27,28,32,30)(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=ghi,c^2=ghi,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8247085);