#8110494

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

3994 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=ghi,e^2=i,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, 8017690);