#6576527

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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	#35687	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2665 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)(7,12)(8,10)(9,11)(13,14,17,21)(15,23,27,19)(16,28,31,20)(18,22,32,39)(24,35,25,34)(26,36,30,38)(29,40,33,37)(41,43,42,44);(5,7,9,8)(6,10,11,12)(15,24)(16,29)(18,32)(19,34)(20,37)(22,39)(23,35)(25,27)(28,40)(31,33)(41,42)(43,44);(2,4)(13,15)(14,19)(16,30)(17,27)(18,25)(20,38)(21,23)(22,35)(24,32)(26,31)(28,36)(29,42)(33,41)(34,39)(37,44)(40,43);(5,8,9,7)(6,10,11,12)(15,25)(16,29)(18,32)(19,35)(20,37)(22,39)(23,34)(24,27)(26,30)(28,40)(31,33)(36,38);(13,16,17,31)(14,20,21,28)(15,26,27,30)(18,33,32,29)(19,36,23,38)(22,40,39,37)(24,41,25,42)(34,43,35,44)

Polycyclic presentation:

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

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, 6483723);