#417385

order	512
nilpotency class	3
isoclinism	isoclinic class of #394189
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

830 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ef,d^2=gh,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324581);