#189657

order	512
nilpotency class	3
isoclinism	isoclinic class of #19504
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#380
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

815 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:

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

Polycyclic presentation:

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

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, 96853);