#7709533

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4348 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)(10,15)(13,23)(16,24)(17,20)(25,26,31,35)(27,34,29,32)(28,30,39,33)(36,38,37,40);(1,2)(3,6)(4,5)(7,8)(9,10,14,15)(11,16,21,24)(12,17,22,20)(13,18,23,19)(25,27,31,29)(26,32,35,34)(28,37,39,36)(30,38,33,40);(10,15)(12,22)(16,24)(18,19)(25,28,31,39)(26,33,35,30)(27,36,29,37)(32,38,34,40);(1,3,4,7)(2,6,5,8)(9,11,14,21)(10,16,15,24)(12,19,22,18)(13,20,23,17)(25,28)(26,30)(27,37)(29,36)(31,39)(32,38)(33,35)(34,40);(9,12)(10,17)(11,19)(13,16)(14,22)(15,20)(18,21)(23,24)(25,29,31,27)(26,34,35,32)(28,36,39,37)(30,40,33,38);(9,13)(10,18)(11,20)(12,16)(14,23)(15,19)(17,21)(22,24)(25,30)(26,28)(27,38)(29,40)(31,33)(32,37)(34,36)(35,39)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=gh,e^2=h,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, 7616729);