#6444612

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

1984 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351808);