#121063

order	512
nilpotency class	3
isoclinism	isoclinic class of #10456
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

560 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)(5,6,9,7,8,12,10,11)(13,14,17,19,18,22,16,21)(15,20,25,26,23,28,24,27);(5,7,9,12,8,11,10,6)(14,19)(15,23)(16,17)(20,27)(21,22)(26,28);(6,11)(7,12)(9,10)(13,15,16,24,18,23,17,25)(14,20,21,27,22,28,19,26)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=di,d^2=h,e^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,c^b=cf,d^b=dh,g^b=gh,e^c=ei,f^c=fhi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28259);