#135992

order	512
nilpotency class	4
isoclinism	isoclinic class of #12114
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1595
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8853
prime radical	C₂⁵	Q₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

438 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 84 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dg,e^b=ehi,g^b=gi,d^c=dh,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227810
maximal normal subgroup of	#72227810

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43188);