#498000

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, prime radical	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

562 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fi,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, 405196);