#333001

order	512
nilpotency class	3
isoclinism	isoclinic class of #332983
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18912	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

763 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240197);