#6446680

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×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×#3499	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4038 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)(7,11)(8,9)(10,12)(13,14,19,24)(15,25,30,37)(16,21,34,42)(17,23,18,22)(20,29,38,28)(26,39,27,43)(31,35,44,36)(32,41,33,40);(1,2,3,4)(6,9)(10,11)(13,15,17,28,19,30,18,29)(14,20,22,25,24,38,23,37)(16,31,33,26,34,44,32,27)(21,39,41,35,42,43,40,36);(5,7,8,12)(6,10,9,11)(13,16)(14,21)(15,26)(17,32)(18,33)(19,34)(20,35)(22,40)(23,41)(24,42)(25,39)(27,30)(28,31)(29,44)(36,38)(37,43);(13,16)(14,21)(15,27)(17,32)(18,33)(19,34)(20,36)(22,40)(23,41)(24,42)(25,43)(26,30)(28,44)(29,31)(35,38)(37,39);(13,17,19,18)(14,22,24,23)(15,28,30,29)(16,32,34,33)(20,25,38,37)(21,40,42,41)(26,31,27,44)(35,39,36,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,f^c=fi,h^c=hi,f^d=fi,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, 6353876);