#8434325

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	#36607	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4504 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^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, 8341521);