#8932275

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

4160 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)(3,8)(4,5)(6,14)(7,9)(10,15)(11,13)(12,16)(17,18,23,26)(19,24,27,31)(20,30,29,22)(21,25,28,32)(33,34)(35,36)(37,45)(38,41)(39,43)(40,46)(42,47)(44,48);(1,3,7,4)(2,8,9,5)(6,11,15,12)(10,16,14,13)(17,19,23,27)(18,24,26,31)(20,28,29,21)(22,25,30,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(2,9)(3,4)(6,15)(13,16)(19,27)(20,29)(22,30)(24,31)(33,34,38,41)(35,39,43,36)(37,45,46,40)(42,44,48,47);(1,4,7,3)(2,8,9,5)(6,12,15,11)(10,16,14,13)(17,20)(18,22)(19,28)(21,27)(23,29)(24,25)(26,30)(31,32)(33,34)(35,39)(36,43)(37,40)(38,41)(42,47)(44,48)(45,46);(1,5,7,8)(2,4,9,3)(6,13,15,16)(10,12,14,11)(17,21,23,28)(18,25,26,32)(19,29,27,20)(22,31,30,24)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,16)(9,14)(17,22)(18,20)(19,25)(21,31)(23,30)(24,28)(26,29)(27,32)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=g,d^2=gh,e^2=gi,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, 8839471);