#8110515

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4076 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)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29)(33,34)(35,43)(36,47)(37,48)(38,42)(39,40)(41,44)(45,46);(2,4)(7,9)(8,12)(14,15)(19,26)(20,28)(22,25)(23,30)(34,40)(35,44)(36,48)(42,46);(1,2,6,4)(3,7,11,9)(5,8,13,12)(10,14,16,15)(17,19)(18,22)(20,27)(21,26)(23,29)(24,25)(28,32)(30,31)(33,35,39,44)(34,41,40,43)(36,45,48,38)(37,46,47,42);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)(33,36)(34,37)(35,45)(38,44)(39,48)(40,47)(41,46)(42,43);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,19)(18,22)(20,27)(21,26)(23,29)(24,25)(28,32)(30,31)(33,37)(34,36)(35,46)(38,43)(39,47)(40,48)(41,45)(42,44);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(33,38,39,45)(34,42,40,46)(35,36,44,48)(37,43,47,41)

Polycyclic presentation:

2^9|a^2=gh,c^2=ghi,d^2=gi,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, 8017711);