#9508641

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	#43032	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4209 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9415837);