#6467496

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2276 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)(13,14,20,37,19,24,18,25)(15,26,36,21,30,42,29,40)(16,31,35,23,33,22,17,34)(27,39,28,44,43,41,32,38);(2,4)(13,15)(14,21)(16,32)(17,27)(18,36)(19,30)(20,29)(22,41)(23,38)(24,40)(25,42)(26,37)(28,33)(31,39)(34,44)(35,43);(6,9)(10,12)(13,16,20,35,19,33,18,17)(14,22,37,34,24,31,25,23)(15,27,36,28,30,43,29,32)(21,38,42,39,40,44,26,41);(13,17,18,33,19,35,20,16)(14,23,25,31,24,34,37,22)(15,28,29,27,30,32,36,43)(21,39,26,38,40,41,42,44);(5,7,8,11)(6,10,9,12)(14,24)(21,40)(22,31)(23,34)(25,37)(26,42)(38,44)(39,41)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374692);