#8110611

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	#59233	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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)(18,23)(20,28)(21,24)(27,32)(33,34,39,43)(35,37,46,40)(36,41,48,45)(38,44,47,42);(2,4)(7,9)(8,12)(14,15)(18,23)(19,26)(20,28)(29,31)(33,35,39,46)(34,40,43,37)(36,44,48,42)(38,45,47,41);(1,2,6,4)(3,7,11,9)(5,8,13,12)(10,14,16,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,36,39,48)(34,41,43,45)(35,44,46,42)(37,47,40,38);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,19,22,26)(18,24,23,21)(20,27,28,32)(25,31,30,29)(33,37)(34,35)(36,47)(38,48)(39,40)(41,44)(42,45)(43,46);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,20,22,28)(18,25,23,30)(19,27,26,32)(21,29,24,31)(33,35)(34,37)(36,44)(38,45)(39,46)(40,43)(41,47)(42,48);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,21,22,24)(18,26,23,19)(20,29,28,31)(25,32,30,27)(33,38,39,47)(34,42,43,44)(35,45,46,41)(36,37,48,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=i,e^2=i,f^2=gi,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, 8017807);