#462445

order	512
nilpotency class	3
isoclinism	isoclinic class of #455564
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

879 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)(11,15)(12,17,24,35)(13,26)(14,19)(16,21)(18,29)(20,38,23,30)(22,32)(25,37,28,36)(27,31)(33,40,34,39);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,26)(17,30)(18,31)(19,32)(23,24)(25,39)(27,29)(28,40)(33,36)(34,37)(35,38);(3,7)(5,8)(9,11)(10,16)(12,23)(13,27)(14,22)(15,26)(17,34)(18,32)(19,31)(20,24)(21,29)(25,39)(28,40)(30,37)(33,35)(36,38);(9,12,13,28,14,24,29,25)(10,17,18,37,19,35,26,36)(11,20,21,40,22,23,27,39)(15,30,31,34,32,38,16,33)

Polycyclic presentation:

2^9|a^2=eg,d^2=ef,f^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,h^c=hi,e^d=ei,f^d=fi,h^d=hi

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, 369641);