#162622

order	512
nilpotency class	3
isoclinism	isoclinic class of #162550
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	#2856	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1008 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)(9,10,12,16,13,14,11,15)(17,18)(19,27,30,23)(20,24)(21,25)(22,26)(28,32,29,31);(1,2)(3,6)(4,8)(5,7)(10,14)(11,12)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,32)(26,27);(9,11,13,12)(10,15,14,16)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,31,27,32);(1,3,5,4)(2,6,7,8)(10,14)(15,16)(17,21,22,20)(18,25,26,24)(19,29,30,28)(23,32,27,31)

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=ghi,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, 69818);