#8260832

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	#56402	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4074 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,d^2=g,e^2=i,f^2=i,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, 8168028);