#161458

order	512
nilpotency class	3
isoclinism	isoclinic class of #161458
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	C₂²×#121	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1438 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)(12,14)(15,16)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,3)(2,5)(4,8)(6,7)(10,13)(11,16)(14,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(19,25)(22,28)(26,31)(29,32);(2,6)(5,7)(10,14)(13,15)

Polycyclic presentation:

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

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, 68654);