#102168

order	512
nilpotency class	3
isoclinism	isoclinic class of #102164
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

494 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)(3,5,7,8)(4,6)(10,13,14,15)(17,18)(19,22)(20,24)(21,23);(1,3)(2,5,6,8)(4,7)(9,10,11,15,12,14,16,13)(17,18,22,24,21,23,20,19);(2,6)(5,8)(17,19,22,18,21,24,20,23)

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=eg,d^2=gh,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=ci,d^b=dh,f^b=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9364);