#209644

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

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

914 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

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, 116840);