#498601

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

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×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

574 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)(6,10)(11,12)(13,14)(15,24)(16,20)(17,30)(18,23)(19,27)(21,25)(22,29)(26,37)(28,39)(31,35)(32,40)(33,44)(34,43)(36,41)(38,42);(1,3,4,2)(5,6)(7,11)(8,12)(9,10);(5,7,9,8)(6,11,10,12)(13,15,25,40,16,26,31,43,18,28,41,44,29,42,17,27)(14,19,30,38,22,33,36,39,23,34,35,37,20,32,21,24);(5,8,9,7)(6,11,10,12)(13,16,18,29)(14,20,23,22)(15,26,28,42)(17,25,31,41)(19,32,34,33)(21,35,36,30)(24,37,39,38)(27,40,43,44)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=efh,d^2=ei,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, 405797);