#8340035

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

4133 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=hi,e^2=g,f^2=hi,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, 8247231);