#6575169

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#36018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2173 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)(13,14,18,23)(15,25,28,20)(16,21,31,34)(17,22,35,32)(19,24,36,41)(26,43,40,33)(27,44,39,30)(29,42,37,38);(6,9)(10,12)(13,15,18,28)(14,20,23,25)(16,30,31,44)(17,33,35,43)(19,37,36,29)(21,39,34,27)(22,40,32,26)(24,42,41,38);(2,4)(13,16,18,31)(14,21,23,34)(15,26,28,40)(17,24,35,41)(19,22,36,32)(20,33,25,43)(27,42,39,38)(29,44,37,30);(5,7,8,11)(6,10,9,12)(16,22)(17,34)(19,36)(21,35)(24,41)(26,44)(27,33)(29,37)(30,40)(31,32)(38,42)(39,43);(13,17,18,35)(14,22,23,32)(15,27,28,39)(16,24,31,41)(19,21,36,34)(20,30,25,44)(26,42,40,38)(29,43,37,33)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=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, 6482365);