#7649181

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4442 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)(3,7,8,5)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,23)(21,22)(25,26,30,33)(27,31,34,38)(28,29,32,37)(35,36,39,40);(1,3,4,8)(2,5,6,7)(25,27)(26,31)(28,35)(29,36)(30,34)(32,39)(33,38)(37,40);(10,14)(15,17)(16,22)(23,24)(27,34)(31,38)(35,39)(36,40);(9,11)(10,15)(12,20)(13,19)(14,17)(16,24)(18,21)(22,23)(25,28)(26,29)(27,35)(30,32)(31,36)(33,37)(34,39)(38,40);(9,12,13,21)(10,16,14,22)(11,18,19,20)(15,23,17,24)(25,29)(26,32)(27,36)(28,33)(30,37)(31,39)(34,40)(35,38);(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,24)(21,22)(25,27)(26,31)(28,35)(29,36)(30,34)(32,39)(33,38)(37,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,e^2=h,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, 7556377);