#498048

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

618 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405244);