#8425017

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4324 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,e^2=h,f^2=gh,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, 8332213);