#163878

order	512
nilpotency class	3
isoclinism	isoclinic class of #17603
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#537	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,7,8,5)(4,6)(9,10,12,15,13,16,11,14)(18,22,24,27)(23,29,30,32);(1,3,4,8)(2,5,6,7)(17,18)(19,23)(20,27)(21,24)(22,28)(25,30)(26,32)(29,31);(9,11,13,12)(10,14,16,15)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31);(19,25)(23,30)(26,31)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 71074);