#6466861

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35329	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2250 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=ghi,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374057);