#118502

order	512
nilpotency class	3
isoclinism	isoclinic class of #10508
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25698);