#8421822

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4487 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8329018);