#149295

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₈×D₄
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

506 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,f^b=fi,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56491);