#7664857

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

5034 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,e^2=gh,f^2=gh,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, 7572053);