#6460050

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	#60453	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)(9,10,14,20,16,22,15,21)(11,23,35,17,26,36,25,37)(12,27,30,18,31,24,29,38)(13,19,32,39,34,40,33,28);(2,5)(6,8)(9,11)(10,17)(12,28)(13,18)(14,25)(15,35)(16,26)(19,29)(20,23)(21,36)(22,37)(24,33)(27,32)(30,40)(31,39)(34,38);(9,12)(10,18,22,38)(11,19,26,40)(13,23)(14,30)(15,29)(16,31)(17,32)(20,24,21,27)(25,28,35,39)(33,37)(34,36);(1,3,4,7)(2,6,5,8)(9,13,15,33,16,34,14,32)(10,19,21,28,22,40,20,39)(11,24,25,18,26,27,35,38)(12,23,29,37,31,36,30,17);(9,14,16,15)(10,20,22,21)(11,25,26,35)(12,29,31,30)(13,32,34,33)(17,23,37,36)(18,27,38,24)(19,39,40,28)

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,d^2=f,e^2=i,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, 6367246);