#8810216

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	#56414	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8717412);