#8096691

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4395 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)(3,7)(4,5)(6,8)(10,15)(11,17)(18,23)(19,24)(25,26,30,33)(27,34,37,31)(28,29,32,38)(35,40,39,36);(2,5)(6,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,24)(22,23)(25,27,30,37)(26,31,33,34)(28,35,32,39)(29,36,38,40);(1,2)(3,6)(4,5)(7,8)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40);(1,3,4,8)(2,6,5,7)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,21,23,20)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40);(9,12,14,16)(10,17,15,11)(13,21,22,20)(18,19,23,24)(25,28)(26,29)(27,35)(30,32)(31,36)(33,38)(34,40)(37,39);(9,13,14,22)(10,18,15,23)(11,19,17,24)(12,20,16,21)(25,29)(26,32)(27,36)(28,33)(30,38)(31,39)(34,35)(37,40)

Polycyclic presentation:

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

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, 8003887);