#183875

order	512
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

(5,6,8,11,9,7,10,12)(13,14)(15,19)(16,18)(17,21)(20,24)(22,26)(23,25)(27,28);(1,2,4,3)(14,18)(16,24)(19,25)(20,21)(23,28)(26,27);(1,3,4,2)(13,15,17,22)(14,19,21,26)(16,23,24,28)(18,25,20,27);(5,7,8,12,9,6,10,11)(15,22)(19,26)(23,28)(25,27)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,d^2=f,e^2=h,f^2=i,b^a=be,e^a=eh,e^b=eh,d^c=dh

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91071);