#120979

order	512
nilpotency class	3
isoclinism	isoclinic class of #120978
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

435 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28175);