#109850

order	512
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#389
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 17046);