#8119578

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

3861 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026774);