#109187

order	512
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

774 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ei,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,c^b=ch,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16383);