#8422262

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4222 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,e^2=h,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, 8329458);