#6440502

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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

2805 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347698);