#6579224

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1981 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,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, 6486420);