#7657740

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	#37620	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4300 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)(12,20)(15,22)(18,24)(21,23)(25,26,31,29)(27,34,32,28)(30,33,38,37)(35,40,39,36);(1,2,5,6)(3,7,4,8)(9,10)(11,16)(12,15)(13,17)(14,19)(18,23)(20,22)(21,24)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,40)(38,39);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,28,31,34)(26,32,29,27)(30,36,38,40)(33,39,37,35);(9,12,13,20)(10,15,17,22)(11,18,19,24)(14,21,16,23);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,29,31,26)(27,34,32,28)(30,33,38,37)(35,36,39,40);(1,4,5,3)(2,8,6,7)(9,10)(11,14)(12,15)(13,17)(16,19)(18,21)(20,22)(23,24)(25,30,31,38)(26,33,29,37)(27,35,32,39)(28,36,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7564936);