#498452

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×D₁₆	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

624 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)(13,14,16,21)(15,24,25,19)(17,20,28,36)(18,31,29,23)(22,38,37,30)(26,32,41,35)(27,34,42,40)(33,44,39,43);(1,3,4,2)(5,6,9,10)(7,11,8,12)(13,15)(14,19)(16,25)(17,26)(18,27)(20,35)(21,24)(22,33)(23,34)(28,41)(29,42)(30,44)(31,40)(32,36)(37,39)(38,43);(5,7,9,8)(6,11,10,12)(14,20)(15,25)(17,30)(18,29)(19,32)(21,36)(22,23)(24,35)(26,43)(28,38)(31,37)(33,40)(34,39)(41,44);(5,8,9,7)(6,11,10,12)(15,25)(19,24)(26,41)(27,42)(32,35)(33,39)(34,40)(43,44)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ei,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,f^c=fh,h^c=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, 405648);