#8928768

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

3988 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)(3,8,11,16)(4,14,12,6)(5,9,15,13)(17,18,23,25)(19,24,27,32)(20,28,29,26)(21,22,30,31)(34,38)(35,41)(40,43)(42,47);(1,3)(2,8)(4,5)(6,13)(7,11)(9,14)(10,16)(12,15)(17,19)(18,24)(20,21)(22,28)(23,27)(25,32)(26,31)(29,30)(33,34,37,38)(35,39,41,45)(36,43,44,40)(42,48,47,46);(3,11)(4,12)(6,14)(8,16)(18,25)(19,27)(20,29)(22,31)(33,34)(35,39)(36,40)(37,38)(41,45)(42,46)(43,44)(47,48);(1,4)(2,6)(3,5)(7,12)(8,13)(9,16)(10,14)(11,15)(17,20,23,29)(18,26,25,28)(19,21,27,30)(22,24,31,32)(33,35)(34,39)(36,42)(37,41)(38,45)(40,46)(43,48)(44,47);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,13)(14,16)(17,21)(18,22)(19,20)(23,30)(24,28)(25,31)(26,32)(27,29)(33,36,37,44)(34,40,38,43)(35,42,41,47)(39,46,45,48);(1,6)(2,4)(3,13)(5,8)(7,14)(9,11)(10,12)(15,16)(17,22)(18,21)(19,28)(20,24)(23,31)(25,30)(26,27)(29,32)(33,35)(34,39)(36,42)(37,41)(38,45)(40,46)(43,48)(44,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=gh,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, 8835964);