#414377

order	512
nilpotency class	3
isoclinism	isoclinic class of #412862
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2858
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15170
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

923 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 923 conjugacy classes of subgroups — too many to draw. Only its 192 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321573);