#7712406

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	#37379	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4407 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)(11,15)(12,14)(13,20)(16,21)(17,18)(19,24)(22,23)(25,26,31,28)(27,35,36,32)(29,33,39,38)(30,37,40,34);(1,3,5,8)(2,6,4,7)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,27)(26,32)(28,35)(29,37)(30,33)(31,36)(34,39)(38,40);(10,12)(11,17)(16,20)(19,23)(27,36)(29,39)(32,35)(33,38);(1,4,5,2)(3,7,8,6)(9,11,14,17)(10,15,12,18)(13,19,21,23)(16,22,20,24)(25,28,31,26)(27,35,36,32)(29,38,39,33)(30,37,40,34);(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,29)(26,33)(27,37)(28,38)(30,32)(31,39)(34,36)(35,40);(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)(25,30,31,40)(26,34,28,37)(27,33,36,38)(29,32,39,35)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=g,e^2=h,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, 7619602);