#8928617

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

4066 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)(3,6)(4,13)(5,9)(7,10)(8,12)(11,16)(14,15)(17,18)(19,21)(20,29)(22,25)(23,24)(26,28)(27,31)(30,32)(33,34)(35,38)(36,41)(37,42)(39,43)(40,45)(44,47)(46,48);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(17,19)(18,21)(20,30)(22,27)(23,28)(24,26)(25,31)(29,32)(33,35)(34,40)(36,42)(37,44)(38,43)(39,45)(41,48)(46,47);(3,11)(4,12)(6,16)(8,13)(17,20,24,22)(18,25,23,29)(19,27,26,30)(21,32,28,31)(33,36,39,47)(34,41,43,44)(35,42,45,46)(37,40,48,38);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,21,24,28)(18,26,23,19)(20,31,22,32)(25,27,29,30)(33,37,39,48)(34,42,43,46)(35,44,45,41)(36,38,47,40);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,22,24,20)(18,25,23,29)(19,27,26,30)(21,31,28,32)(33,38)(34,35)(36,37)(39,40)(41,42)(43,45)(44,46)(47,48);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,23,24,18)(19,28,26,21)(20,25,22,29)(27,32,30,31)(33,35)(34,38)(36,46)(37,44)(39,45)(40,43)(41,48)(42,47)

Polycyclic presentation:

2^9|c^2=gh,d^2=gh,e^2=h,f^2=h,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, 8835813);