#8459108

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

4592 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=h,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8366304);