#193812

order	512
nilpotency class	3
isoclinism	isoclinic class of #29610
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1138 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)(3,11,13,15)(4,14,7,8)(5,9)(6,10)(12,16)(17,18,20,24,21,23,27,30)(19,22,26,29,25,28,31,32);(1,3,5,14,6,13,16,8)(2,7,9,15,10,4,12,11)(17,19,20,26,21,25,27,31)(18,22,24,29,23,28,30,32);(1,4,6,7)(2,8,10,14)(3,12,13,9)(5,11,16,15)(19,25)(22,28)(26,31)(29,32);(18,23)(22,28)(24,30)(29,32)

Polycyclic presentation:

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

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, 101008);