#7647714

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4349 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=h,f^2=hi,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, 7554910);