#6467486

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2244 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)(13,14,17,24,18,23,19,25)(15,26,28,37,29,38,30,20)(16,22,32,41,33,21,34,40)(27,39,43,35,44,42,31,36);(2,4)(13,15,18,29)(14,20,23,37)(16,31,33,43)(17,30,19,28)(21,39,22,42)(24,38,25,26)(27,32,44,34)(35,40,36,41);(6,9)(10,12)(13,16,17,32,18,33,19,34)(14,21,24,40,23,22,25,41)(15,27,28,43,29,44,30,31)(20,35,26,42,37,36,38,39);(13,16,17,32,18,33,19,34)(14,22,24,41,23,21,25,40)(15,27,28,43,29,44,30,31)(20,36,26,39,37,35,38,42);(5,7,8,11)(6,10,9,12)(14,23)(20,37)(21,22)(24,25)(26,38)(35,36)(39,42)(40,41)

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,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, 6374682);