#6457709

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444406
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60454	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1988 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6364905);