#97641

order	512
nilpotency class	3
isoclinism	isoclinic class of #8412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

774 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=ghi,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,f^b=fgh,e^c=eg,f^c=fg

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4837);