#139522

order	512
nilpotency class	4
isoclinism	isoclinic class of #139432
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 652 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=gh,d^2=ghi,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=fhi,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	#72238678, #72238680, #72238683
maximal normal subgroup of	#72238678, #72238680, #72238683

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46718);