#448519

order	512
nilpotency class	3
isoclinism	isoclinic class of #417575
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1021 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(2,6)(7,8)(10,13)(14,15)(17,18)(19,26)(20,28,30,31)(21,24)(22,23)(25,27,32,29);(1,2,5,6)(3,7,4,8)(9,10,12,13)(11,15,16,14)(17,19,21,22)(18,23,24,26)(20,29,30,27)(25,28,32,31);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(18,24)(19,22)(25,32)(27,29);(1,4,5,3)(2,7,6,8)(10,13)(14,15)(17,20)(18,25)(19,27)(21,30)(22,29)(23,31)(24,32)(26,28)

Polycyclic presentation:

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

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, 355715);