#6404611

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
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	C₂²×D₈	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1829 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)(8,11)(9,12)(13,14)(15,23)(16,27)(17,21)(18,32)(19,25)(20,30)(22,31)(24,38)(26,34)(28,42)(29,41)(33,43)(35,39)(36,44)(37,40);(5,7,10,6)(8,11,9,12)(13,15,17,25)(14,19,21,23)(16,28,29,24)(18,26,31,40)(20,35,36,33)(22,34,32,37)(27,38,41,42)(30,43,44,39);(13,16,18,30,17,29,31,44)(14,20,22,27,21,36,32,41)(15,24,26,39,25,28,40,43)(19,33,34,42,23,35,37,38);(5,8,10,9)(6,11,7,12)(15,25)(19,23)(24,28)(26,40)(33,35)(34,37)(38,42)(39,43);(1,3,4,2)(5,9,10,8)(6,11,7,12)(14,21)(15,25)(16,29)(22,32)(26,40)(30,44)(33,35)(38,42)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6311807);