#242862

order	512
nilpotency class	3
isoclinism	isoclinic class of #233019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

906 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)(3,6)(4,5)(7,8)(9,10,13,15,11,14,12,16)(17,18)(19,22)(20,24)(21,23)(25,29)(26,31)(27,30)(28,32);(2,5)(6,8)(10,14)(15,16)(17,19)(18,22)(20,25)(21,27)(23,30)(24,29)(26,28)(31,32);(1,3,4,7)(2,6,5,8)(10,15)(12,13)(14,16)(19,25)(20,28)(22,29)(24,32)(26,27)(30,31);(18,23)(22,30)(24,32)(29,31)

Polycyclic presentation:

2^9|a^2=fh,c^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,f^c=fh,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, 150058);