#6467490

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	#25785	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2316 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)(13,14,18,25,20,24,19,26)(15,27,30,39,32,40,31,21)(16,22,34,23,35,41,17,36)(28,43,29,42,44,38,33,37);(2,4)(13,15)(14,21)(16,33)(17,28)(18,31)(19,30)(20,32)(22,38)(23,42)(24,39)(25,40)(26,27)(29,35)(34,44)(36,37)(41,43);(6,9)(10,12)(13,16,18,34,20,35,19,17)(14,22,25,23,24,41,26,36)(15,28,30,29,32,44,31,33)(21,37,27,43,39,42,40,38);(13,17,19,35,20,34,18,16)(14,23,26,22,24,36,25,41)(15,29,31,28,32,33,30,44)(21,38,40,42,39,43,27,37);(5,7,8,11)(6,10,9,12)(14,24)(21,39)(22,41)(23,36)(25,26)(27,40)(37,42)(38,43)

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=h,e^2=g,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, 6374686);