#115132

order	512
nilpotency class	3
isoclinism	isoclinic class of #115124
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

989 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 989 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=hi,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,e^c=ehi,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22328);