#98043

order	512
nilpotency class	3
isoclinism	isoclinic class of #97894
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

647 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=h,e^2=g,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=eg

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5239);