#194353

order	512
nilpotency class	3
isoclinism	isoclinic class of #18810
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1077 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fi,g^c=gi

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, 101549);