#162620

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

1040 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)(3,5)(4,7)(6,8)(10,14,15,16)(17,18,20,24,21,23,29,31)(19,25,27,32,26,22,28,30);(2,5)(3,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,28)(21,26)(23,25)(24,32)(27,29)(30,31);(19,26)(22,25)(27,28)(30,32);(9,11,13,12)(10,14,15,16)(18,23)(22,25)(24,31)(30,32)

Polycyclic presentation:

2^9|a^2=f,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, 69816);