#498489

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂×#2215	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

670 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,4,3)(6,9)(10,12)(13,14,18,22)(15,24,28,40)(16,20,30,23)(17,31,32,34)(19,33,36,27)(21,25,38,29)(26,39,44,41)(35,42,37,43);(1,3,4,2)(5,6)(7,10)(8,9)(11,12);(5,7,8,11)(6,10,9,12)(13,15,25,42,16,26,17,27,18,28,29,43,30,44,32,33)(14,19,34,39,23,37,21,24,22,36,31,41,20,35,38,40);(6,9)(10,12)(13,16,18,30)(14,20,22,23)(15,26,28,44)(17,29,32,25)(19,35,36,37)(21,34,38,31)(24,39,40,41)(27,43,33,42)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=efhi,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=de

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, 405685);