#315307

order	512
nilpotency class	3
isoclinism	isoclinic class of #288787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

652 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)(9,10,14,11,12,15,13,16)(17,18)(19,29)(20,27)(21,24)(22,25)(23,28)(26,31)(30,32);(1,3,4,7)(2,5,6,8)(9,11,14,15,12,16,13,10)(17,19,20,30,23,31,22,21)(18,24,25,26,28,32,27,29);(2,6)(5,8)(10,11)(13,14)(15,16)(17,20,23,22)(18,25,28,27)(19,30,31,21)(24,26,32,29);(10,15)(11,16)(17,21,22,31,23,30,20,19)(18,26,27,24,28,29,25,32)

Polycyclic presentation:

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

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, 222503);