#7712634

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	#36596	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4776 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=gi,e^2=gh,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, 7619830);