#8115856

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4278 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8023052);