#7664601

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	#38949	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5383 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

(1,2,4,6)(3,5,8,7)(9,10,13,16)(11,14,12,15)(17,18,20,22)(19,23,24,21);(9,10)(11,14)(12,15)(13,16)(19,24)(21,23);(1,2)(3,7)(4,6)(5,8)(17,19)(18,21)(20,24)(22,23);(9,11,13,12)(10,14,16,15)(17,18)(19,21)(20,22)(23,24);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,16,15);(1,2)(3,7)(4,6)(5,8)(9,12,13,11)(10,15,16,14)

Polycyclic presentation:

2^9|a^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fhi,c^b=ci,e^c=ehi,f^e=fhi

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, 7571797);