#227159

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
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)(3,9,13,16,14,15,12,7)(17,18)(19,29)(20,30)(21,25)(22,27)(23,26)(24,28)(31,32);(1,3,4,12,6,14,5,13)(2,7,8,15,11,16,10,9)(17,19,24,31)(18,20,28,21)(22,25,23,30)(26,32,27,29);(1,4,6,5)(2,8,11,10)(3,12,14,13)(7,15,16,9)(17,20,24,21)(18,19,28,31)(22,29,23,32)(25,26,30,27);(1,3,6,14)(2,9,11,15)(4,12,5,13)(7,10,16,8)(17,21,24,20)(18,25,28,30)(19,26,31,27)(22,29,23,32)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,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=dei,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, 134355);