#8108730

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
prime radical	#40198	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4076 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015926);