#6370214

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	C₂⁵	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6277410);