#8339968

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	#58258	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4157 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)(3,5)(4,12)(6,7)(8,10)(9,15)(11,16)(13,14)(17,18,22,25)(19,26,20,23)(21,24,29,31)(27,32,28,30)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,47)(44,48);(2,7)(3,10)(9,12)(11,13)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,31,32)(35,36)(39,42)(43,44)(47,48);(2,7)(5,8)(9,12)(14,16)(21,29)(24,31)(27,28)(30,32)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,3)(2,8)(4,13)(5,7)(6,10)(9,14)(11,15)(12,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,30)(28,32)(29,31)(33,36,38,35)(34,39,41,42)(37,43,46,44)(40,48,45,47);(1,4,6,15)(2,9,7,12)(3,11,10,13)(5,14,8,16)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,31,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48);(1,5,6,8)(2,3,7,10)(4,14,15,16)(9,11,12,13)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=i,d^2=i,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8247164);