#227585

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	#2358
prime radical	#18510	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

673 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134781);