#8897974

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

4453 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)(3,4)(5,6)(7,8)(25,26,32,30)(27,33,31,38)(28,36,29,34)(35,40,37,39);(2,6)(3,7)(9,10,14,15)(11,16,21,24)(12,17,22,20)(13,18,23,19)(25,27,32,31)(26,33,30,38)(28,37,29,35)(34,40,36,39);(2,6)(4,8)(10,15)(13,23)(16,24)(17,20)(25,28,32,29)(26,34,30,36)(27,35,31,37)(33,39,38,40);(2,6)(3,7)(9,11)(10,16)(12,18)(13,20)(14,21)(15,24)(17,23)(19,22)(25,29,32,28)(26,34,30,36)(27,35,31,37)(33,40,38,39);(1,3,5,7)(2,4,6,8)(9,12)(10,17)(11,19)(13,16)(14,22)(15,20)(18,21)(23,24)(25,30,32,26)(27,33,31,38)(28,36,29,34)(35,39,37,40);(1,4)(2,7)(3,6)(5,8)(9,13)(10,18)(11,20)(12,16)(14,23)(15,19)(17,21)(22,24)(25,31,32,27)(26,33,30,38)(28,35,29,37)(34,40,36,39)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8805170);