#8424915

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4534 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)(11,17)(12,18)(13,16)(14,15)(19,23)(20,24)(21,22)(26,31)(32,33)(34,37)(39,40);(1,2)(3,8)(4,6)(5,7)(10,15)(11,18)(16,22)(19,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,3,5,6)(2,4,7,8)(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,24)(22,23)(25,27,30,28)(26,32,31,33)(29,35,38,36)(34,39,37,40);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,24)(22,23)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,40)(36,39);(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,28,30,27)(26,33,31,32)(29,36,38,35)(34,40,37,39);(1,4)(2,6)(3,7)(5,8)(9,13,14,21)(10,16,15,22)(11,19,18,24)(12,20,17,23)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40)

Polycyclic presentation:

2^9|c^2=gi,e^2=gh,f^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, 8332111);