#6466181

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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2200 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)(7,11)(8,9)(10,12)(13,14,18,24,19,23,33,41)(15,25,28,37,29,42,34,20)(16,21,30,39,31,38,17,22)(26,36,27,40,32,44,43,35);(2,4)(6,9)(10,11)(13,15,19,29)(14,20,23,37)(16,27,31,43)(17,32,30,26)(18,34,33,28)(21,36,38,44)(22,40,39,35)(24,42,41,25);(5,7,8,12)(6,10,9,11)(13,16,18,30,19,31,33,17)(14,21,24,39,23,38,41,22)(15,26,28,27,29,32,34,43)(20,35,25,36,37,40,42,44);(13,17,33,31,19,30,18,16)(14,22,41,38,23,39,24,21)(15,27,34,26,29,43,28,32)(20,36,42,35,37,44,25,40);(14,23)(20,37)(21,38)(22,39)(24,41)(25,42)(35,40)(36,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=gh,d^2=hi,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, 6373377);