#116923

order	512
nilpotency class	3
isoclinism	isoclinic class of #116918
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

623 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 623 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=h,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,e^b=ei,e^c=ehi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24119);