#6517088

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500416
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	#35329	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2238 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6424284);