#6465978

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444408
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1810 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,18)(13,19)(14,17)(15,20)(16,24)(22,32)(23,37)(25,31)(26,40)(27,36)(28,33)(29,35)(30,39)(34,38);(2,5)(6,8)(9,11)(10,16)(12,26)(13,24)(14,25)(15,22)(17,32)(18,34)(19,21)(20,31)(23,29)(27,38)(28,39)(30,35)(33,37)(36,40);(10,17,19,20)(11,22,24,25)(12,27)(18,35,36,33)(23,26,39,38)(28,29)(30,37)(34,40);(9,12,15,29,13,27,14,28)(10,18,20,35,19,36,17,33)(11,23,25,26,24,39,22,38)(16,30,32,34,21,37,31,40);(1,3,4,7)(2,6,5,8)(12,27)(18,36)(23,39)(26,38)(28,29)(30,37)(33,35)(34,40)

Polycyclic presentation:

2^9|c^2=gh,d^2=h,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,d^c=di,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, 6373174);