#227583

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	#18494	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

648 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,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, 134779);