#315171

order	512
nilpotency class	3
isoclinism	isoclinic class of #288755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

606 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)(3,5)(4,6)(7,8)(10,13)(11,14)(15,16)(17,18,22,28,23,26,21,27)(19,24,30,32,20,31,29,25);(1,3,4,7)(2,5,6,8)(9,10,14,16,12,15,11,13)(17,19,21,29,23,20,22,30)(18,24,27,25,26,31,28,32);(2,6)(5,8)(17,20,23,19)(18,25,26,32)(21,29,22,30)(24,28,31,27);(9,10,14,16,12,15,11,13)(18,26)(24,31)(25,32)(27,28)

Polycyclic presentation:

2^9|a^2=g,b^2=efh,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,f^c=fi,g^c=gi

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, 222367);