#6404144

order	512
nilpotency class	3
isoclinism	isoclinic class of #62265
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3269	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2096 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)(7,10)(8,9)(11,12)(13,14)(15,20)(16,26)(17,22)(18,33)(19,23)(21,28)(24,36)(25,34)(27,41)(29,42)(30,43)(31,38)(32,44)(35,40)(37,39);(6,9)(7,11)(13,15,19,18)(14,20,23,33)(16,24,29,28)(17,25,31,32)(21,26,36,42)(22,34,38,44)(27,39,30,40)(35,41,37,43);(13,16,15,24,19,29,18,28)(14,21,33,42,23,36,20,26)(17,30,25,40,31,27,32,39)(22,37,44,41,38,35,34,43);(5,7,8,11)(6,10,9,12)(13,17)(14,22)(15,25)(16,27)(18,32)(19,31)(20,34)(21,35)(23,38)(24,39)(26,41)(28,40)(29,30)(33,44)(36,37)(42,43);(1,3,4,2)(6,9)(10,12)(17,31)(22,38)(25,32)(27,30)(34,44)(35,37)(39,40)(41,43)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6311340);