#7652362

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

4201 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=gi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

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, 7559558);