#8096506

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3867 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003702);