#6448912

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2268 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)(15,23)(16,28)(17,21)(18,22)(19,26)(20,30)(24,34)(25,36)(27,35)(29,37)(31,40)(32,39)(33,38)(41,42)(43,44);(2,4)(13,15)(14,19)(16,29)(17,26)(18,25)(20,37)(21,23)(22,35)(24,31)(27,32)(28,44)(30,43)(33,42)(34,38)(36,39)(40,41);(6,9)(10,12)(13,16,17,30)(14,20,21,28)(15,24,26,42)(18,33,32,31)(19,34,23,41)(22,40,39,38)(25,29,27,43)(35,37,36,44);(5,7,8,11)(6,10,9,12)(15,25)(18,32)(19,35)(22,39)(23,36)(24,29)(26,27)(31,33)(34,37)(38,40)(41,44)(42,43);(14,21)(15,26)(20,28)(22,39)(24,42)(25,27)(29,43)(38,40)

Polycyclic presentation:

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

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, 6356108);