#8388208

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4327 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8295404);