#6440522

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

2802 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)(15,25)(16,21)(17,22)(18,23)(19,24)(20,28)(26,37)(27,36)(29,44)(30,41)(31,40)(32,39)(33,42)(34,43)(35,38);(2,4)(5,7,8,11)(6,9,10,12)(13,15)(14,20)(16,30)(17,32)(18,28)(19,35)(21,39)(22,41)(23,25)(24,44)(26,31)(27,33)(29,34)(36,40)(37,42)(38,43);(6,10)(9,12)(13,16,18,17)(14,21,23,22)(15,26,28,27)(19,33,34,31)(20,36,25,37)(24,42,43,40)(29,32,35,30)(38,41,44,39);(13,17,18,16)(14,22,23,21)(15,27,28,26)(19,31,34,33)(20,37,25,36)(24,40,43,42)(29,30,35,32)(38,39,44,41);(14,23)(15,28)(21,22)(24,43)(26,27)(29,35)(30,32)(40,42)

Polycyclic presentation:

2^9|a^2=fi,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=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, 6347718);