#243308

order	512
nilpotency class	3
isoclinism	isoclinic class of #10260
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#121
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ 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)(3,6)(4,5)(7,8)(9,10,11,14,12,15,16,13)(17,18)(19,22)(20,23)(21,24)(25,32)(26,30)(27,29)(28,31);(2,5)(6,8)(17,19,20,26)(18,22,23,30)(21,25,28,27)(24,32,31,29);(10,13)(11,16)(14,15)(18,23)(19,25)(21,28)(22,29)(26,27)(30,32);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150504);