#8127343

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	#54179	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4083 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)(17,18,22,26)(19,27,30,23)(20,31,29,24)(21,25,28,32)(33,34,39,36)(35,42,46,40)(37,47,45,41)(38,48,44,43);(2,6)(7,9)(8,12)(14,15)(17,19,22,30)(18,23,26,27)(20,21,29,28)(24,25,31,32)(33,35)(34,40)(36,42)(37,44)(38,45)(39,46)(41,43)(47,48);(2,6)(3,10)(8,12)(11,16)(17,20,22,29)(18,24,26,31)(19,28,30,21)(23,32,27,25)(33,36,39,34)(35,42,46,40)(37,41,45,47)(38,48,44,43);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,32)(25,31)(26,27)(33,37,39,45)(34,41,36,47)(35,44,46,38)(40,43,42,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,32)(29,31)(33,35)(34,42)(36,40)(37,44)(38,45)(39,46)(41,48)(43,47);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,31,27,24)(33,38,39,44)(34,43,36,48)(35,45,46,37)(40,41,42,47)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,d^2=gh,f^2=g,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, 8034539);