#340416

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
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
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

698 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)(7,10)(8,11)(9,12)(13,14,19,24)(15,25,26,17)(16,23,18,21)(20,28,27,22);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,19,26)(14,20,24,27)(16,22,18,28)(17,21,25,23);(2,4)(6,11)(7,8)(10,12)(13,16,19,18)(14,21,24,23)(15,22,26,28)(17,20,25,27);(6,11)(7,8)(10,12)(13,17,16,27,19,25,18,20)(14,22,21,15,24,28,23,26)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cg,d^b=dei,e^b=eh,d^c=dh,f^c=fi,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, 247612);