#139483

order	512
nilpotency class	4
isoclinism	isoclinic class of #139432
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

670 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=i,d^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fh,g^b=gi,h^b=hi,d^c=dgh,f^c=fgh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251909, #72252005, #72252006
maximal normal subgroup of	#72251909, #72252005, #72252006

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46679);