#6443834

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2110 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,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, 6351030);