#497991

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
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×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

594 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)(7,10)(8,12)(9,11)(13,14)(15,24)(16,29)(17,20)(18,21)(19,26)(22,30)(23,31)(25,40)(27,35)(28,44)(32,38)(33,39)(34,42)(36,41)(37,43);(5,7,9,8)(6,10,11,12)(13,15,17,27)(14,19,20,34)(16,25,32,41)(18,33,31,28)(21,36,23,40)(22,39,37,44)(24,38,35,29)(26,43,42,30);(5,8,9,7)(6,10,11,12)(14,20)(19,34)(21,23)(22,37)(24,35)(29,38)(36,40)(39,44);(14,21)(15,25)(16,30)(18,31)(19,35)(20,23)(24,34)(26,28)(27,41)(29,38)(32,43)(33,42)(36,39)(40,44)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,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

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405187);