#7658969

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

4088 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

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, 7566165);