#6464064

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1834 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:

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

Polycyclic presentation:

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

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, 6371260);