#7678306

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×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4822 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)(3,6)(4,8)(5,7)(9,10,13,12)(11,15,16,14)(17,18,23,21)(19,26,20,24)(22,29,30,25)(27,31,28,32);(1,3,5,8)(2,6,7,4)(9,11,13,16)(10,14,12,15)(17,19,23,20)(18,24,21,26)(22,27,30,28)(25,31,29,32);(1,4)(2,3)(5,6)(7,8)(9,10,13,12)(11,14,16,15)(18,21)(19,20)(25,29)(27,28);(1,2)(3,6)(4,8)(5,7)(9,12,13,10)(11,15,16,14)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32);(17,21,23,18)(19,24,20,26)(22,29,30,25)(27,31,28,32);(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7585502);