#111297

order	512
nilpotency class	3
isoclinism	isoclinic class of #10524
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

796 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 796 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2)(3,4)(6,10,11,12)(13,14,18,23,19,22,17,24)(15,25,28,37,29,20,27,36)(16,30,33,21,32,39,34,40)(26,38,31,41,42,44,43,35);(2,4)(5,6)(7,12)(8,10)(9,11)(13,15,19,29)(14,20,22,25)(16,31,32,43)(17,28,18,27)(21,38,40,44)(23,37,24,36)(26,34,42,33)(30,41,39,35);(5,6)(7,10)(8,12)(9,11)(13,16,19,32)(14,21,22,40)(15,26,29,42)(17,34,18,33)(20,35,25,41)(23,39,24,30)(27,43,28,31)(36,38,37,44)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,e^2=hi,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=dh,e^c=ehi,f^c=fh,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, 18493);