#7649954

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	#38437	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4403 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,e^2=g,f^2=gh,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, 7557150);