#7712233

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4333 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=gh,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, 7619429);