#314721

order	512
nilpotency class	3
isoclinism	isoclinic class of #262731
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	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

619 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 221917);