#333934

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

849 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

(5,6,9,11,10,8,7,12)(13,14,19,22,18,16,17,21)(15,23,26,20,27,28,25,24);(1,2)(3,4)(5,7)(6,8)(9,10)(13,15)(14,20)(16,24)(17,26)(18,27)(19,25)(21,28)(22,23);(2,4)(5,7,10,9)(6,11,8,12)(13,16,18,14)(15,24,27,20)(17,22,19,21)(23,25,28,26);(5,8,10,6)(7,12,9,11)(15,25)(17,19)(20,23)(21,22)(24,28)(26,27)

Polycyclic presentation:

2^9|a^2=ei,c^2=h,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=dei,e^b=eh,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 241130);