#406018

order	512
nilpotency class	3
isoclinism	isoclinic class of #406018
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14922
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1065 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313214);