#7660699

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

4260 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7567895);