#471581

order	512
nilpotency class	3
isoclinism	isoclinic class of #455107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1181 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:

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

Polycyclic presentation:

2^9|b^2=egh,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, 378777);