#6467504

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2259 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)(13,14,18,25,19,24,20,26)(15,27,30,39,31,40,32,21)(16,33,35,41,36,22,17,23)(28,38,29,37,44,43,34,42);(2,4)(13,15)(14,21)(16,34)(17,28)(18,32)(19,31)(20,30)(22,38)(23,42)(24,39)(25,40)(26,27)(29,36)(33,43)(35,44)(37,41);(6,9)(10,12)(13,16,20,17,19,36,18,35)(14,22,26,41,24,33,25,23)(15,28,32,34,31,44,30,29)(21,37,40,38,39,42,27,43);(13,17,20,36,19,35,18,16)(14,23,26,22,24,41,25,33)(15,29,32,28,31,34,30,44)(21,38,40,42,39,43,27,37);(5,7,8,11)(6,10,9,12)(14,24)(21,39)(22,33)(23,41)(25,26)(27,40)(37,42)(38,43)

Polycyclic presentation:

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

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, 6374700);