#226209

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

522 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=fh,d^2=h,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, 133405);