#221092

order	512
nilpotency class	3
isoclinism	isoclinic class of #20286
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	#3475	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

875 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=chi,d^b=de,e^b=eh,f^c=fi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 128288);