#8455729

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4311 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)(3,4)(5,7)(6,8)(9,10,16,14)(11,21,22,17)(12,19,13,18)(15,20,23,24)(25,26,30,29)(27,33,34,31)(28,32,36,37)(35,39,40,38);(1,3)(2,6)(4,7)(5,8)(9,11,16,22)(10,17,14,21)(12,15,13,23)(18,20,19,24)(27,34)(28,36)(31,33)(32,37);(9,12,16,13)(10,18,14,19)(11,15,22,23)(17,20,21,24)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39);(2,7)(4,6)(9,13,16,12)(10,19,14,18)(11,15,22,23)(17,20,21,24)(25,28,30,36)(26,32,29,37)(27,35,34,40)(31,38,33,39);(1,3)(2,6)(4,7)(5,8)(9,14,16,10)(11,21,22,17)(12,19,13,18)(15,24,23,20);(1,4,5,6)(2,3,7,8)(9,15)(10,20)(11,13)(12,22)(14,24)(16,23)(17,19)(18,21)(25,29,30,26)(27,33,34,31)(28,32,36,37)(35,38,40,39)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=h,e^2=hi,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, 8362925);