#7712302

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4364 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^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, 7619498);