#8824726

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	#47955	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4393 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)(3,6)(4,8)(5,7)(11,17)(15,18)(19,23)(22,24)(25,26,31,29)(27,28,34,32)(30,37,38,33)(35,39,40,36);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(26,29)(27,34)(33,37)(35,40);(1,3)(2,4)(5,8)(6,7)(10,14)(15,18)(16,20)(22,24)(25,27,31,34)(26,32,29,28)(30,35,38,40)(33,39,37,36);(1,3,5,8)(2,6,7,4)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28)(26,27)(29,34)(30,36)(31,32)(33,35)(37,40)(38,39);(1,3)(2,4)(5,8)(6,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,29,31,26)(27,28,34,32)(30,37,38,33)(35,36,40,39);(1,4)(2,3)(5,6)(7,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,30,31,38)(26,33,29,37)(27,35,34,40)(28,36,32,39)

Polycyclic presentation:

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

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, 8731922);