#7649932

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	#40515	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4491 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)(9,10,14,17)(11,15,18,22)(12,13,16,21)(19,20,23,24)(26,31)(27,32)(33,37)(34,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,26,30,31)(27,28,32,36)(29,33,38,37)(34,35,39,40);(11,18)(15,22)(19,23)(20,24)(25,26,30,31)(27,28,32,36)(29,37,38,33)(34,40,39,35);(9,12)(10,13)(11,19)(14,16)(15,20)(17,21)(18,23)(22,24)(25,27)(26,28)(29,34)(30,32)(31,36)(33,35)(37,40)(38,39);(9,13)(10,16)(11,20)(12,17)(14,21)(15,23)(18,24)(19,22)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,37)(38,40);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,29)(26,33)(27,34)(28,35)(30,38)(31,37)(32,39)(36,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,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, 7557128);