#187624

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2228 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 2228 conjugacy classes of subgroups — too many to draw. Only its 322 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 — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,f^c=fh,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94820);