#440799

order	512
nilpotency class	3
isoclinism	isoclinic class of #417575
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1176 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,c^2=ei,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, 347995);