#8932303

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

4168 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839499);