#8271170

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4055 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)(3,5)(4,12)(6,7)(8,10)(9,13)(11,15)(14,16)(17,18,22,21)(19,23,27,25)(20,28,29,24)(26,31,32,30)(33,34,39,40)(35,41,36,45)(37,42,48,38)(43,44,46,47);(2,7)(3,10)(4,13)(14,15)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(34,40)(37,48)(41,45)(43,46);(2,7)(5,8)(9,12)(14,15)(18,21)(23,25)(24,28)(30,31)(33,35,39,36)(34,41,40,45)(37,43,48,46)(38,47,42,44);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,36,39,35)(34,41,40,45)(37,46,48,43)(38,47,42,44);(1,4,6,13)(2,9,7,12)(3,11,10,16)(5,14,8,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37)(34,42)(35,43)(36,46)(38,40)(39,48)(41,44)(45,47);(1,5,6,8)(2,3,7,10)(4,14,13,15)(9,11,12,16)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30)(33,38,39,42)(34,37,40,48)(35,44,36,47)(41,46,45,43)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,d^2=ghi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8178366);