#289203

order	512
nilpotency class	3
isoclinism	isoclinic class of #289187
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25404	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

639 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,d^2=ei,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gi,d^c=dh,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196399);