#7715509

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

4081 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622705);