#112174

order	512
nilpotency class	3
isoclinism	isoclinic class of #112171
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

781 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 781 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=dhi,f^c=fh,g^c=gh

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, 19370);