#139824

order	512
nilpotency class	4
isoclinism	isoclinic class of #139782
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

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

normal subgroups only

This group has 624 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) = 32

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

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

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,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

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268087, #72268088
maximal normal subgroup of	#72268087, #72268088

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47020);