#404898

order	512
nilpotency class	3
isoclinism	isoclinic class of #394048
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

1159 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312094);