#6483412

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2098 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)(17,24)(18,23)(19,22)(20,28)(21,33)(26,35)(27,37)(29,36)(31,38)(32,39)(34,40)(41,43)(42,44);(1,2,3,4)(7,8)(10,12)(13,15,17,28,19,27,18,29)(14,20,23,25,22,36,24,37)(16,31,32,43,34,26,33,42)(21,38,40,44,39,35,30,41);(5,7,9,8)(6,10,11,12)(13,16,18,33,19,34,17,32)(14,21,24,30,22,39,23,40)(15,26,29,43,27,31,28,42)(20,35,37,44,36,38,25,41);(5,8,9,7)(6,10,11,12)(15,27)(20,36)(25,37)(26,31)(28,29)(35,38)(41,44)(42,43);(6,11)(10,12)(14,22)(15,27)(21,39)(23,24)(26,31)(28,29)(30,40)(42,43)

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=hi,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^b=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, 6390608);