#194604

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂³×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1450 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1450 conjugacy classes of subgroups — too many to draw. Only its 258 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) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=i,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,f^a=fh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fhi,g^c=gi

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, 101800);