#6483474

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2086 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=g,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

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, 6390670);