#410065

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^c=fi

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, 317261);