#7673644

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,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, 7580840);