#6454495

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1955 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)(15,20)(16,21)(17,19)(18,22)(23,36)(24,35)(25,31)(26,38)(27,32)(28,37)(29,33)(30,34)(39,44)(40,42)(41,43);(2,4)(14,19)(15,23)(16,26)(18,29)(20,34)(21,37)(24,40)(25,30)(27,28)(31,36)(32,38)(43,44);(6,9)(10,12)(13,15,17,25)(14,20,19,31)(16,24,27,40)(18,30,29,23)(21,35,32,42)(22,34,33,36)(26,41,28,39)(37,44,38,43);(13,16,17,27)(14,21,19,32)(15,24,25,40)(18,26,29,28)(20,35,31,42)(22,38,33,37)(23,39,30,41)(34,43,36,44);(5,7,8,11)(6,10,9,12)(15,25)(20,31)(23,30)(24,40)(34,36)(35,42)(39,41)(43,44)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361691);