#8110434

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	#36828	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=hi,e^2=hi,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, 8017630);