#8932278

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

3987 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=h,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, 8839474);