#6467499

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2276 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=hi,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi

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, 6374695);