#205896

order	512
nilpotency class	3
isoclinism	isoclinic class of #9784
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

929 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,d^2=hi,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,c^b=ch,e^b=ei,g^b=gi

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, 113092);