#6370438

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	C₂×#532	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2352 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=g,h^2=i,b^a=bf,c^a=ci,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, 6277634);