#196108

order	512
nilpotency class	3
isoclinism	isoclinic class of #196101
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#447	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1209 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103304);