#8745828

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

3952 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)(3,10)(4,8)(5,7)(6,9)(11,16)(12,15)(13,14)(17,18,23,20)(19,27,29,24)(21,30,28,25)(22,26,32,31)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,14,15,16)(17,19,23,29)(18,24,20,27)(21,22,28,32)(25,26,30,31)(34,35)(39,41)(40,42)(46,47);(4,12)(8,15)(11,13)(14,16)(18,20)(24,27)(25,30)(26,31)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,20,23,18)(19,24,29,27)(21,30,28,25)(22,26,32,31)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,21)(18,25)(19,22)(20,30)(23,28)(24,26)(27,31)(29,32);(1,5,6,3)(2,7,9,10)(4,11,12,13)(8,16,15,14)(17,22)(18,26)(19,28)(20,31)(21,29)(23,32)(24,30)(25,27)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=i,d^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653024);