#462316

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,h^b=hi,d^c=de,f^c=fi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 369512);