#139825

order	512
nilpotency class	4
isoclinism	isoclinic class of #139783
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1889
commutator	C₂²×Q₈	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	#3486	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

570 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=gh,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,h^a=hi,c^b=ci,d^b=dgi,e^b=ehi,f^b=fgi,d^c=dh,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268326, #72268328
maximal normal subgroup of	#72268326, #72268328

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47021);