#8211667

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4061 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=ghi,e^2=gh,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, 8118863);