#414435

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

1477 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 1477 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)(4,18)(5,10)(6,11)(7,12)(8,15)(9,21)(14,23)(16,25)(17,24)(19,26)(20,28)(22,27)(29,32)(30,31);(1,3,5,15,7,17,6,16)(2,8,11,13,12,25,10,24)(4,19,20,30,22,29,21,14)(9,26,27,23,28,32,18,31);(1,4,5,21,7,22,6,20)(2,9,11,18,12,28,10,27)(3,14,16,19,17,30,15,29)(8,23,24,32,25,31,13,26);(2,10,12,11)(8,24,25,13)(9,27,28,18)(23,32,31,26)

Polycyclic presentation:

2^9|b^2=e,c^2=f,d^2=gh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 321631);