#9267035

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4058 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gi,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9174231);