#105325

order	512
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#95
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₄×#95
prime radical	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=hi,g^2=hi,b^a=bd,d^a=dh,f^a=fhi,e^b=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12521);