#7712314

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	#36360	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4517 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)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26,31,28)(27,35,36,32)(29,33,38,39)(30,40,37,34);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27)(26,32)(28,35)(29,37)(30,38)(31,36)(33,40)(34,39);(10,14)(11,18)(16,20)(19,24)(27,36)(29,38)(32,35)(33,39);(1,2,5,4)(3,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28,31,26)(27,35,36,32)(29,39,38,33)(30,40,37,34);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,29,31,38)(26,33,28,39)(27,37,36,30)(32,40,35,34);(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,30,31,37)(26,34,28,40)(27,38,36,29)(32,39,35,33)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619510);