#464277

order	512
nilpotency class	3
isoclinism	isoclinic class of #455107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1454 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371473);