#225776

order	512
nilpotency class	3
isoclinism	isoclinic class of #19883
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	D₄×C₄⋊C₄
prime radical	C₂×#135	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

535 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,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,e^d=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 132972);