#8197162

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	#47685	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8104358);