#231519

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
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	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

642 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138715);