#7644254

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

4081 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)(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23)(25,26,31,34)(27,35,30,32)(28,38,33,29)(36,40,39,37);(1,2)(3,7)(4,8)(5,6)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27,31,30)(26,32,34,35)(28,39,33,36)(29,37,38,40);(10,12)(15,17)(16,20)(22,23)(25,28,31,33)(26,29,34,38)(27,36,30,39)(32,40,35,37);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,29)(26,33)(27,37)(28,34)(30,40)(31,38)(32,36)(35,39);(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,30,31,27)(26,32,34,35)(28,39,33,36)(29,40,38,37)

Polycyclic presentation:

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

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, 7551450);