#414448

order	512
nilpotency class	3
isoclinism	isoclinic class of #413183
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2867
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15172
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1525 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 1525 conjugacy classes of subgroups — too many to draw. Only its 240 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)(3,13)(4,9)(5,10)(6,11)(7,12)(8,15)(14,29)(16,23)(17,24)(18,22)(19,27)(20,26)(21,28)(25,30)(31,32);(1,3,5,15,7,16,6,17)(2,8,11,13,12,24,10,23)(4,18,19,30,21,31,20,14)(9,25,26,22,28,29,27,32);(1,4)(2,9)(3,14,16,30)(5,20)(6,19)(7,21)(8,22,24,32)(10,26)(11,27)(12,28)(13,25,23,29)(15,31,17,18);(2,10,12,11)(8,23,24,13)(9,26,28,27)(22,29,32,25)

Polycyclic presentation:

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

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, 321644);