#8927244

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

4146 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=h,e^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, 8834440);