#6453968

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35210	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3098 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)(13,14,19,23)(15,26,29,39)(16,31,17,22)(18,25,20,24)(21,30,38,28)(27,43,44,40)(32,37,35,36)(33,41,34,42);(2,4)(13,15)(14,21)(16,32)(17,35)(18,30)(19,29)(20,28)(22,40)(23,38)(24,39)(25,26)(27,34)(31,43)(33,44)(36,42)(37,41);(6,9)(10,12)(13,16,19,17)(14,22,23,31)(15,27,29,44)(18,34,20,33)(21,36,38,37)(24,42,25,41)(26,40,39,43)(28,32,30,35);(13,17,19,16)(14,22,23,31)(15,27,29,44)(18,33,20,34)(21,37,38,36)(24,42,25,41)(26,43,39,40)(28,32,30,35);(5,7,8,11)(6,10,9,12)(14,23)(21,38)(22,31)(24,25)(26,39)(36,37)(40,43)(41,42)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,e^2=gi,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^b=fi,h^b=hi,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, 6361164);