#7655735

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4366 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=gh,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, 7562931);