#497724

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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

702 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=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, 404920);