#162584

order	512
nilpotency class	3
isoclinism	isoclinic class of #162550
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 69780);