#10225563

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	#45917	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4281 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:

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

Polycyclic presentation:

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

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, 10132759);