#8728690

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

3943 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)(3,5)(4,11)(6,7)(8,13)(9,15)(10,12)(14,16)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32)(34,38)(39,44)(40,45)(46,48);(2,7)(3,9)(8,11)(10,16)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32)(35,41)(39,44)(42,47)(46,48);(2,7)(3,9)(8,11)(10,16)(19,26)(22,23)(27,32)(30,31)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,3,6,9)(2,5,7,15)(4,10,13,16)(8,14,11,12)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,4,6,13)(2,8,7,11)(3,10,9,16)(5,14,15,12)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,5)(2,3)(4,12)(6,15)(7,9)(8,16)(10,11)(13,14)(17,18)(19,22)(20,28)(21,25)(23,26)(24,29)(27,31)(30,32)(33,36,37,43)(34,40,38,45)(35,42,41,47)(39,46,44,48)

Polycyclic presentation:

2^9|d^2=g,e^2=g,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, 8635886);