#8757416

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3767 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=g,e^2=hi,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, 8664612);