#139499

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

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

normal subgroups only

This group has 690 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=ghi,d^2=gh,f^2=ghi,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	#72238675, #72238678, #72238679
maximal normal subgroup of	#72238675, #72238678, #72238679

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46695);