#160922

order	512
nilpotency class	3
isoclinism	isoclinic class of #160892
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

858 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:

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

Polycyclic presentation:

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

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, 68118);