#8922828

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

4217 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)(3,10)(4,8)(5,13)(6,7)(9,14)(11,12)(15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,31)(30,32)(33,34,38,43)(35,41,46,39)(36,37,42,40)(44,48,47,45);(2,7)(4,10)(9,13)(12,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,48)(41,43)(42,47);(2,7)(3,8)(5,14)(12,16)(18,22)(20,29)(23,25)(26,32)(36,42)(37,40)(44,47)(45,48);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,36,38,42)(34,40,43,37)(35,44,46,47)(39,48,41,45);(1,4)(2,8)(3,7)(5,12)(6,10)(9,15)(11,13)(14,16)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,35)(34,41)(36,47)(37,45)(38,46)(39,43)(40,48)(42,44);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,37,38,40)(34,42,43,36)(35,45,46,48)(39,47,41,44)

Polycyclic presentation:

2^9|a^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, 8830024);