#8360935

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	#35406	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8268131);