#8188572

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095768);