#6441646

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2214 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,20)(17,22)(18,21)(23,32)(24,33)(25,35)(26,34)(27,36)(28,38)(29,37)(30,40)(31,39)(41,43)(42,44);(2,3)(5,7,8,11)(6,9,10,12)(13,15)(14,19)(16,27)(17,24)(18,25)(20,36)(21,35)(22,33)(23,28)(26,31)(29,42)(30,41)(32,38)(34,39)(37,44)(40,43);(1,2,4,3)(6,10)(9,12)(13,16,18,29)(14,20,21,37)(15,23,25,41)(17,30,31,28)(19,32,35,43)(22,40,39,38)(24,27,26,42)(33,36,34,44);(1,3,4,2)(15,24)(17,31)(19,33)(22,39)(23,27)(25,26)(28,30)(32,36)(34,35)(38,40)(41,42)(43,44);(14,21)(15,25)(20,37)(22,39)(23,41)(24,26)(27,42)(38,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fi,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,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, 6348842);