#117825

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

918 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 25021);