#243229

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

Aliases

No aliases for this group.

Subgroups

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

696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,8)(4,6)(5,7)(9,10,13,14,12,16,11,15)(17,18)(19,22)(20,24)(21,23)(25,29)(26,31)(27,30)(28,32);(1,3,5,4)(2,6,7,8)(17,19)(18,22)(20,25)(21,27)(23,30)(24,29)(26,28)(31,32);(1,4,5,3)(2,6,7,8)(10,14)(11,13)(15,16)(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,b^2=e,c^2=e,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, 150425);