#9838175

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

4402 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)(17,20)(19,24)(25,26,30,33)(27,34,29,31)(28,36,38,32)(35,39,37,40);(1,2)(3,7)(4,5)(6,8)(9,10)(11,16)(12,20)(13,18)(14,15)(17,22)(19,23)(21,24)(25,27,30,29)(26,31,33,34)(28,35,38,37)(32,39,36,40);(1,3)(2,6)(4,7)(5,8)(25,28,30,38)(26,32,33,36)(27,35,29,37)(31,39,34,40);(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,21,20,23)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,36)(34,40);(1,4,5,2)(3,7,8,6)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,23,16,21)(25,29,30,27)(26,31,33,34)(28,37,38,35)(32,39,36,40);(1,3)(2,7)(4,6)(5,8)(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,19,20,24)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9745371);