#305234

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

737 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=eh,c^2=efh,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=di,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212430);