#7711903

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

4741 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)(9,10,13,16)(11,14,12,15)(17,18)(19,23)(20,29)(21,31)(22,24)(25,30)(26,28)(27,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19,22,27)(18,23,24,32)(20,28,30,21)(25,31,29,26);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(18,24)(19,27)(20,30)(26,31);(9,11,13,12)(10,14,16,15)(17,18)(19,23)(20,25)(21,26)(22,24)(27,32)(28,31)(29,30);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,15,16,14)(17,20)(18,25)(19,28)(21,27)(22,30)(23,31)(24,29)(26,32);(1,3,4,7)(2,6,5,8)(17,21,22,28)(18,26,24,31)(19,20,27,30)(23,25,32,29)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=g,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, 7619099);