#315043

order	512
nilpotency class	3
isoclinism	isoclinic class of #288715
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	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

662 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)(17,18,21,25,20,24,27,23)(19,26,30,32,29,22,28,31);(1,3,4,7)(2,5,6,8)(9,10,14,16,12,15,11,13)(17,19,27,28,20,29,21,30)(18,22,23,32,24,26,25,31);(2,6)(5,8)(18,23)(19,28)(21,27)(22,26)(24,25)(29,30);(9,10,14,16,12,15,11,13)(18,24)(22,26)(23,25)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 222239);