#332856

order	512
nilpotency class	3
isoclinism	isoclinic class of #332839
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

779 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)(9,10,15,19,16,11,14,18)(12,21,24,13,17,22,23,20);(1,3,4,5)(2,6,8,7)(9,11,16,10)(12,22,17,21)(13,23,20,24)(14,18,15,19);(2,6)(3,5)(7,8)(9,12,16,17)(10,13,11,20)(14,23,15,24)(18,21,19,22);(2,7)(3,5)(6,8)(9,13)(10,17)(11,12)(14,22)(15,21)(16,20)(18,23)(19,24)

Polycyclic presentation:

2^9|a^2=eh,b^2=i,c^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=def,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240052);