#8317518

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

4082 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=gi,e^2=gi,f^2=g,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, 8224714);