#339727

order	512
nilpotency class	3
isoclinism	isoclinic class of #339703
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3151	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

615 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=gh,f^c=fi,g^c=gh,f^d=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, 246923);