#8754824

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

3991 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)(3,9)(4,12)(5,10)(6,7)(8,14)(11,13)(15,16)(19,25)(23,26)(27,31)(30,32)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(2,7)(5,9)(8,12)(13,15)(17,18)(19,26)(20,28)(21,22)(23,25)(24,29)(27,30)(31,32)(33,34,39,36)(35,40,38,42)(37,44,46,41)(43,48,45,47);(2,7)(3,10)(8,12)(11,16)(18,22)(23,26)(24,28)(30,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,15)(14,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,36,39,34)(35,42,38,40)(37,44,46,41)(43,48,45,47);(1,4,6,14)(2,8,7,12)(3,11,10,16)(5,13,9,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,37,39,46)(34,41,36,44)(35,43,38,45)(40,47,42,48);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,16)(14,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8662020);