#6460053

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444408
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	#60454	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,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^d=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, 6367249);