#6457708

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	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1932 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,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, 6364904);