#8455727

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	#40054	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4321 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)(4,5)(6,8)(9,10)(11,18)(12,17)(13,22)(14,16)(15,19)(20,24)(21,23)(25,26,31,28)(27,32,29,33)(30,34,38,36)(35,39,37,40);(2,5)(6,7)(9,11,15,14)(10,16,19,18)(12,20,13,21)(17,23,22,24)(25,27,31,29)(26,32,28,33)(30,35,38,37)(34,39,36,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,13)(10,17,19,22)(11,20,14,21)(16,23,18,24)(25,27,31,29)(26,33,28,32)(30,35,38,37)(34,40,36,39);(1,3,4,8)(2,6,5,7)(9,13,15,12)(10,17,19,22)(11,21,14,20)(16,23,18,24)(25,28,31,26)(27,32,29,33)(30,36,38,34)(35,39,37,40);(9,14,15,11)(10,18,19,16)(12,21,13,20)(17,24,22,23)(25,29,31,27)(26,32,28,33)(30,37,38,35)(34,39,36,40);(9,10)(11,16)(12,17)(13,22)(14,18)(15,19)(20,23)(21,24)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=ghi,d^2=gh,e^2=ghi,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, 8362923);