#118456

order	512
nilpotency class	3
isoclinism	isoclinic class of #118454
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#388
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂×#121	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=ei,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25652);