#7645531

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

4529 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)(10,11)(13,21)(15,17)(19,24)(25,26,30,28)(27,32,35,31)(29,36,37,33)(34,38,40,39);(1,2)(3,6)(4,7)(5,8)(25,27)(26,31)(28,32)(29,34)(30,35)(33,38)(36,39)(37,40);(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,26,30,28)(27,32,35,31)(29,33,37,36)(34,39,40,38);(1,3,5,4)(2,6,8,7)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,28,30,26)(27,32,35,31)(29,36,37,33)(34,39,40,38);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,4,5,3)(2,7,8,6)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,37)(26,33,28,36)(27,34,35,40)(31,38,32,39)

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,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, 7552727);