#6448134

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2062 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fgi,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,e^d=ei

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, 6355330);