#7712030

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

4752 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,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, 7619226);