#226208

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	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

523 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=fh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,c^b=ci,d^b=de,e^b=eh,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, 133404);