#9815296

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

4174 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=ghi,e^2=g,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, 9722492);