#7766336

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	#35005	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4502 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)(9,10,14,15)(11,12,17,20)(13,16,21,22)(18,19,23,24);(1,3,5,8)(2,6,4,7)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24)(25,26,30,31)(27,32,37,40)(28,33,38,36)(29,34,39,35);(1,4,5,2)(3,7,8,6)(10,15)(11,17)(13,21)(19,24)(26,31)(29,39)(32,40)(33,36);(9,11,14,17)(10,12,15,20)(13,18,21,23)(16,19,22,24)(25,27)(26,32)(28,34)(29,36)(30,37)(31,40)(33,39)(35,38);(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,28)(26,33)(27,35)(29,32)(30,38)(31,36)(34,37)(39,40);(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,29)(26,34)(27,36)(28,32)(30,39)(31,35)(33,37)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei

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, 7673532);