#9250225

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

4069 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=i,d^2=h,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, 9157421);