#6467607

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	#25787	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2289 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)(13,14,20,35,19,25,18,24)(15,26,34,21,30,42,29,38)(16,22,17,23,31,39,32,40)(27,37,43,36,33,44,28,41);(2,4)(13,15)(14,21)(16,28)(17,33)(18,34)(19,30)(20,29)(22,37)(23,41)(24,42)(25,38)(26,35)(27,32)(31,43)(36,40)(39,44);(6,9)(10,12)(13,16,18,32,19,31,20,17)(14,22,24,40,25,39,35,23)(15,27,29,28,30,33,34,43)(21,36,26,37,38,41,42,44);(13,17,20,31,19,32,18,16)(14,23,35,39,25,40,24,22)(15,28,34,27,30,43,29,33)(21,37,42,36,38,44,26,41);(5,7,8,11)(6,10,9,12)(16,31)(17,32)(22,39)(23,40)(27,33)(28,43)(36,41)(37,44)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374803);