#8127578

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

4054 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8034774);