#7746751

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4532 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi

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, 7653947);