#8437771

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	#53961	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4429 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8344967);