#231508

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2366
prime radical	#23798	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

718 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,d^2=eh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=de,e^b=eh,g^b=gh,d^c=dhi

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, 138704);