#237334

order	512
nilpotency class	3
isoclinism	isoclinic class of #237325
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1303 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,22,26)(19,27,30,31)(20,25,21,24)(23,29,32,28);(1,3,6,13)(2,7,9,15)(4,12,5,11)(8,16,14,10)(17,19,22,30)(18,23,26,32)(20,29,21,28)(24,31,25,27);(18,24)(19,28)(20,21)(25,26)(27,31)(29,30);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,20,22,21)(18,25,26,24)(19,28,30,29)(23,31,32,27)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144530);