#6465649

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3147	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3154 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=hi,e^2=i,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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372845);