#334796

order	512
nilpotency class	3
isoclinism	isoclinic class of #334695
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

722 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 241992);