#7658626

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	#37553	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4240 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)(15,21)(18,24)(22,23)(25,26)(27,31)(28,37)(29,32)(30,33)(34,40)(35,38)(36,39);(1,2)(3,7)(4,8)(5,6)(9,10)(11,16)(12,21)(13,17)(14,19)(15,20)(18,22)(23,24)(25,27,30,36)(26,31,33,39)(28,34,32,38)(29,35,37,40);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23)(25,28,30,32)(26,29,33,37)(27,34,36,38)(31,35,39,40);(9,12)(10,15)(11,18)(13,20)(14,22)(16,23)(17,21)(19,24)(25,27)(26,31)(28,34)(29,35)(30,36)(32,38)(33,39)(37,40);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,33)(32,37)(34,35)(36,39)(38,40);(1,4,5,3)(2,8,6,7)(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,21)(23,24)(25,29)(26,32)(27,35)(28,33)(30,37)(31,38)(34,39)(36,40)

Polycyclic presentation:

2^9|b^2=h,c^2=h,e^2=g,f^2=g,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, 7565822);