#479284

order	512
nilpotency class	3
isoclinism	isoclinic class of #476659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

850 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,11,16,12,17,10,27)(7,18,20,15,21,29,9,13)(8,22,24,14,26,31,25,32)(19,28,23,36,34,33,35,30);(5,7,11,20,12,21,10,9)(6,13,27,29,17,15,16,18)(8,23,25,19,26,35,24,34)(14,30,31,28,32,36,22,33);(5,8,10,24,12,26,11,25)(6,14,16,22,17,32,27,31)(7,19,9,23,21,34,20,35)(13,28,18,30,15,33,29,36);(2,4)(5,9,11,21,12,20,10,7)(6,15,16,18,17,13,27,29)(8,19,25,23,26,34,24,35)(14,30,22,28,32,36,31,33)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386480);