#8472916

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4709 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

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, 8380112);