#8455746

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4276 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,d^2=gh,e^2=ghi,f^2=gh,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, 8362942);