#475081

order	512
nilpotency class	3
isoclinism	isoclinic class of #455110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×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×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

860 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 382277);