#6460048

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	#60456	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2156 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,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, 6367244);