#8339892

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3969 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:

(1,2,6,9)(3,4,11,7)(5,8,13,15)(10,12,16,14)(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);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(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);(5,13)(8,15)(10,16)(12,14)(18,22)(19,26)(25,29)(27,32)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(1,4,6,7)(2,3,9,11)(5,12,13,14)(8,10,15,16)(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);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,16)(14,15)(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);(1,5,6,13)(2,8,9,15)(3,10,11,16)(4,12,7,14)(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=gi,b^2=hi,c^2=hi,d^2=ghi,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, 8247088);