#433902

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1235 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=fi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341098);