#7709920

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4812 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=h,e^2=gh,f^2=gh,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, 7617116);