#8924722

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

4236 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=ghi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8831918);