#118025

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

670 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,e^b=ehi,d^c=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25221);