#7711935

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

4608 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)(17,18,23,28)(19,26,21,24)(20,25,29,30)(22,32,31,27);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19,23,21)(18,24,28,26)(20,22,29,31)(25,27,30,32);(2,5)(6,8)(17,18)(19,26)(20,30)(21,24)(22,27)(23,28)(25,29)(31,32);(9,11,13,12)(10,14,16,15)(17,20)(18,25)(19,22)(21,31)(23,29)(24,27)(26,32)(28,30);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,15,16,14)(17,21,23,19)(18,26,28,24)(20,31,29,22)(25,32,30,27);(1,3,4,7)(2,6,5,8)(17,22,23,31)(18,27,28,32)(19,29,21,20)(24,30,26,25)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=gh,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, 7619131);