#98217

order	512
nilpotency class	3
isoclinism	isoclinic class of #98134
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

789 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=ghi,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5413);