#6370220

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,g^2=i,h^2=i,b^a=bf,c^a=ci,d^a=df,c^b=cf,e^c=ei,e^d=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277416);