#8934609

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	#60034	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4477 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841805);