#99366

order	512
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

651 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=h,d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dg,e^b=eh,d^c=dh,e^c=ei

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, 6562);