#6466183

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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2238 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)(7,11)(8,9)(10,12)(13,14,18,25,20,26,19,24)(15,27,31,39,32,38,30,21)(16,22,34,41,35,42,17,23)(28,37,29,43,44,40,33,36);(2,4)(6,9)(10,11)(13,15)(14,21)(16,33)(17,28)(18,30)(19,31)(20,32)(22,40)(23,36)(24,27)(25,38)(26,39)(29,35)(34,44)(37,42)(41,43);(5,7,8,12)(6,10,9,11)(13,16,18,34,20,35,19,17)(14,22,25,41,26,42,24,23)(15,28,31,29,32,44,30,33)(21,36,27,37,39,43,38,40);(13,17,19,35,20,34,18,16)(14,23,24,42,26,41,25,22)(15,29,30,28,32,33,31,44)(21,37,38,36,39,40,27,43);(13,18,20,19)(14,24,26,25)(15,30,32,31)(16,34,35,17)(21,27,39,38)(22,23,42,41)(28,33,44,29)(36,37,43,40)

Polycyclic presentation:

2^9|a^2=fg,c^2=ghi,d^2=h,e^2=i,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

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, 6373379);