#8127574

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

4058 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,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, 8034770);