#108251

order	512
nilpotency class	3
isoclinism	isoclinic class of #108238
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

957 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,e^b=ei,f^b=fhi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15447);